This is typical of flat pack furniture—the drawer bottoms are just too thin to hold too much weight, and simply bend/warp over time.

The bottom panel can be put back in place and once straightened should stay in place. Depending how close the bottom piece is to the bottom of the drawer box, you can add a reinforcement bar or two—either below the panel or on top. Think something like a 1x2 piece of lumber, cut to fit inside the drawer box. Use glue and some short screws to screw through the panel into the bar. The bar should keep the panel straighter, and keep it from falling out of the slots in the drawer box.

Edit:

Adding a couple of pics to kinda show what I mean.

This pic shows the bottom of the drawer. In my case the drawer has a slot that tracks a rail in the cabinet box; the piece with the slot had cracked and I had to replace it. In this drawer, the bottom panel was also too small to reach the slots in all the sides, so it wouldn't stay in place consistently. You can see where I put in some screws into the side panel, sticking out to keep the bottom panel from falling out.

Here you can see the screws I put through the bottom panel to fasten it to the track bar beneath. With this bar secured to it, the bottom panel won't flex enough with the weight of its contents to work its way out of the front and rear slots.