Inspired by user6178746's answer, I have the following:

# Given a dict of dataframes, for example:
# dfs = {'gadgets': df_gadgets, 'widgets': df_widgets}

writer = pd.ExcelWriter(filename, engine='xlsxwriter')
for sheetname, df in dfs.items():  # loop through `dict` of dataframes
    df.to_excel(writer, sheet_name=sheetname)  # send df to writer
    worksheet = writer.sheets[sheetname]  # pull worksheet object
    for idx, col in enumerate(df):  # loop through all columns
        series = df[col]
        max_len = max((
            series.astype(str).map(len).max(),  # len of largest item
            len(str(series.name))  # len of column name/header
            )) + 1  # adding a little extra space
        worksheet.set_column(idx, idx, max_len)  # set column width
writer.save()

Try to use this helper function (updated version):

Old version, which is no longer compatible with Pandas 1.3.0+:

import numpy as np
import pandas as pd
from pathlib import Path
from typing import Union, Optional, List, Tuple
from openpyxl import load_workbook
from openpyxl.utils import get_column_letter


def append_df_to_excel(
        filename: Union[str, Path],
        df: pd.DataFrame,
        sheet_name: str = 'Sheet1',
        startrow: int = None,
        max_col_width: int = 40,
        autofilter: bool = False,
        fmt_int: str = "#,##0",
        fmt_float: str = "#,##0.00",
        fmt_date: str = "yyyy-mm-dd",
        fmt_datetime: str = "yyyy-mm-dd hh:mm",
        truncate_sheet: bool = False,
        **to_excel_kwargs
) -> None:
    """
    Append a DataFrame [df] to existing Excel file [filename]
    into [sheet_name] Sheet.
    If [filename] doesn't exist, then this function will create it.

    @param filename: File path or existing ExcelWriter
                     (Example: '/path/to/file.xlsx')
    @param df: DataFrame to save to workbook
    @param sheet_name: Name of sheet which will contain DataFrame.
                       (default: 'Sheet1')
    @param startrow: upper left cell row to dump data frame.
                     Per default (startrow=None) calculate the last row
                     in the existing DF and write to the next row...
    @param max_col_width: maximum column width in Excel. Default: 30
    @param autofilter: boolean - whether add Excel autofilter or not. Default: True
    @param fmt_int: Excel format for integer numbers
    @param fmt_float: Excel format for float numbers
    @param fmt_date: Excel format for dates
    @param fmt_datetime: Excel format for datetime's
    @param truncate_sheet: truncate (remove and recreate) [sheet_name]
                           before writing DataFrame to Excel file
    @param to_excel_kwargs: arguments which will be passed to `DataFrame.to_excel()`
                            [can be a dictionary]
    @return: None

    Usage examples:

    >>> append_df_to_excel('d:/temp/test.xlsx', df, autofilter=True,
                           freeze_panes=(1,0))

    >>> append_df_to_excel('d:/temp/test.xlsx', df, header=None, index=False)

    >>> append_df_to_excel('d:/temp/test.xlsx', df, sheet_name='Sheet2',
                           index=False)

    >>> append_df_to_excel('d:/temp/test.xlsx', df, sheet_name='Sheet2',
                           index=False, startrow=25)

    >>> append_df_to_excel('d:/temp/test.xlsx', df, index=False,
                           fmt_datetime="dd.mm.yyyy hh:mm")

    (c) MaxU
    """
    def set_column_format(ws, column_letter, fmt):
        for cell in ws[column_letter]:
            cell.number_format = fmt
    filename = Path(filename)
    file_exists = filename.is_file()
    # process parameters
    first_col = int(to_excel_kwargs.get("index", True)) + 1
    sheet_name = to_excel_kwargs.get("sheet_name", "Sheet1")
    # ignore [engine] parameter if it was passed
    if 'engine' in to_excel_kwargs:
        to_excel_kwargs.pop('engine')

    with pd.ExcelWriter(
        filename.with_suffix(".xlsx"),
        engine="openpyxl",
        mode="a" if file_exists else "w",
        date_format=fmt_date,
        datetime_format=fmt_datetime,
        **to_excel_kwargs
    ) as writer:
        if file_exists:
            # try to open an existing workbook
            writer.book = load_workbook(filename)
            # get the last row in the existing Excel sheet
            # if it was not specified explicitly
            if startrow is None and sheet_name in writer.book.sheetnames:
                startrow = writer.book[sheet_name].max_row
            # truncate sheet
            if truncate_sheet and sheet_name in writer.book.sheetnames:
                # index of [sheet_name] sheet
                idx = writer.book.sheetnames.index(sheet_name)
                # remove [sheet_name]
                writer.book.remove(writer.book.worksheets[idx])
                # create an empty sheet [sheet_name] using old index
                writer.book.create_sheet(sheet_name, idx)

            # copy existing sheets
            writer.sheets = {ws.title:ws for ws in writer.book.worksheets}
        else:
            # file doesn't exist, we are creating a new one
            startrow = 0

        # write out the DataFrame to an ExcelWriter
        df.to_excel(writer, sheet_name=sheet_name, startrow=startrow,
                    **to_excel_kwargs)

        # automatically set columns' width
        worksheet = writer.sheets[sheet_name]
        for xl_col_no, dtyp in enumerate(df.dtypes, first_col):
            col_no = xl_col_no - first_col
            width = max(df.iloc[:, col_no].astype(str).str.len().max(),
                        len(df.columns[col_no]) + 6)
            width = min(max_col_width, width)
            # print(f"column: [{df.columns[col_no]} ({dtyp.name})]\twidth:\t[{width}]")
            column_letter = get_column_letter(xl_col_no)
            worksheet.column_dimensions[column_letter].width = width
            if np.issubdtype(dtyp, np.integer):
                set_column_format(worksheet, column_letter, fmt_int)
            if np.issubdtype(dtyp, np.floating):
                set_column_format(worksheet, column_letter, fmt_float)
        if autofilter:
            worksheet.auto_filter.ref = worksheet.dimensions

I found the answer in official documentation as @jmcnamara suggested:

    import pandas as pd

    # Create a Pandas dataframe from some data.
    df = pd.DataFrame({'Numbers':    [1010, 2020, 3030, 2020, 1515, 3030, 4545],
                    'Percentage': [.1,   .2,   .33,  .25,  .5,   .75,  .45 ],
    })

    # Create a Pandas Excel writer using XlsxWriter as the engine.
    writer = pd.ExcelWriter("pandas_column_formats.xlsx", engine='xlsxwriter')

    # Convert the dataframe to an XlsxWriter Excel object.
    df.to_excel(writer, sheet_name='Sheet1')

    # Get the xlsxwriter workbook and worksheet objects.
    workbook  = writer.book
    worksheet = writer.sheets['Sheet1']

    # Add some cell formats.
    format1 = workbook.add_format({'num_format': '#,##0.00'})
    format2 = workbook.add_format({'num_format': '0%'})

    # Note: It isn't possible to format any cells that already have a format such
    # as the index or headers or any cells that contain dates or datetimes.

    # Set the column width and format.
    worksheet.set_column('B:B', 18, format1)

    # Set the format but not the column width.
    worksheet.set_column('C:C', None, format2)

The code above is taken from https://xlsxwriter.readthedocs.io/example_pandas_column_formats.html#ex-pandas-column-formats.