There is no "linking" of buttons to functions, nor callback functions.

To do what you're looking for, calling copy when you get a "Copy Button"event back from a Read.

I urge you to read through the docs to get an understanding how these calls to Button, etc, work. http://www.PySimpleGUI.org

Here's what I think you are looking for your code to do:

import PySimpleGUI as sg
import shutil, errno
src = ""
dest = ""
def copy(src, dest):
    try:
        shutil.copytree(src, dest)
    except OSError as e:
        # If the error was caused because the source wasn't a directory
        if e.errno == errno.ENOTDIR:
            shutil.copy(src, dest)
        else:
            print('Directory not copied. Error: %s' % e)

#Me testing out commands in PSG
layout = [[ sg.Text("Select path from source to destination")],
[sg.Text("Source Folder", size=(15,1)), sg.InputText(src),
sg.FolderBrowse()],
[sg.Text("Destination Folder", size=(15,1)),
sg.InputText(dest), sg.FolderBrowse()],
[sg.Button("Transfer", button_color=("white", "blue"), size=
(6, 1)),sg.Button("Copy", button_color=("white",
"green"), size=(6, 1)),sg.Exit(button_color=("white", "red"),
size=(6, 1))]]

window = sg.Window("Mass File Transfer").Layout(layout)

while True:
    event, values = window.Read()
    print(event, values)
    if event in (None, 'Exit'):
        break

    if event == 'Copy':
        copy(values[0], values[1])

The PySimpleGUI documentation discusses how to do this in the section on events / callbacks https://pysimplegui.readthedocs.io/#the-event-loop-callback-functions

It is not lot the other Python GUI frameworks that use callbacks to signal button presses. Instead all button presses are returned as "events" coming back from a Read call.

To achieve a similar result, you check the event and make the function call yourself.

import PySimpleGUI as sg

def func(message):
    print(message)

layout = [[sg.Button('1'), sg.Button('2'), sg.Exit()] ]

window = sg.Window('ORIGINAL').Layout(layout)

while True:             # Event Loop
    event, values = window.Read()
    if event in (None, 'Exit'):
        break
    if event == '1':
        func('Pressed button 1')
    elif event == '2':
        func('Pressed button 2')
window.Close()

To see this code run online, you can run it here using the web version: https://repl.it/@PySimpleGUI/Call-Func-When-Button-Pressed

Added 4/5/2019 I should have also stated in my answer that you could add the event checks to right after your call to Read. You don't have to use an Event Loop as I showed. It could look like this:

event, values = window.Layout(layout).Read()   # from OP's existing code
if event == '1':
    func('Pressed button 1')
elif event == '2':
    func('Pressed button 2')

[ Edit Nov 2020 ] - Callable keys

This isn't a new capability, just didn't mention it in the answer previously.

You can set keys to be functions and then call them when the event is generated. Here is an example that uses a few ways of doing this.

import PySimpleGUI as sg

def func(message='Default message'):
    print(message)

layout = [[sg.Button('1', key=lambda: func('Button 1 pressed')), 
           sg.Button('2', key=func), 
           sg.Button('3'), 
           sg.Exit()]]

window = sg.Window('Window Title', layout)

while True:             # Event Loop
    event, values = window.read()
    if event in (None, 'Exit'):
        break
    if callable(event):
        event()
    elif event == '3':
        func('Button 3 pressed')

window.close()