There are three questions asked in the garb of one title. I've addressed them below.

First, about the title:

How can I find out what's using all of my internal storage?

I use an app called DiskUsage to see what's using my Internal/External Storage. The diagrams (hierarchy in size) it shows are extremely helpful. If your device is rooted, then the possibilities to scan directories in / pops up.

It's simple to use as well. Launch the app, choose the storage and see the stuff. You can also choose to see any folder inside your default File Manager by selecting the folder and tapping on Show.

About the question before the end:

This problem came up when I tried to install an app and the Play Store app told me that the internal memory was full.

This Play Store message is more or less related to the memory available for your apps installation (/data) and not the internal storage which you possibly have understood AFAIK. You can see similar questions/answer under the tag insufficient-memory.

The last question:

How can I free that space?

• If the space belongs to internal storage, then you have multiple options to delete the unnecessary stuff. You can either use DiskUsage which also gives Delete option under menu, or delete the content using your preferred File Manager, or use PC to delete the stuff.
• If the space belongs to /data then you'll have to uninstall the apps.

There are three questions asked in the garb of one title. I've addressed them below.

First, about the title:

How can I find out what's using all of my internal storage?

I use an app called DiskUsage to see what's using my Internal/External SD card. The diagrams (hierarchy in size) it shows are extremely helpful. If your device is rooted, then the possibilities to scan directories in / pops up.

It's simple to use as well. Launch the app, choose the storage and see the stuff. You can also choose to see any folder inside your default File Manager by selecting the folder and tapping on Show.

About the question before the end:

This problem came up when I tried to install an app and the Play Store app told me that the internal memory was full.

This Play Store message is more or less related to the memory available for your apps installation (/data) and not the Internal SD card which you possibly have understood AFAIK. You can see similar questions/answer under the tag insufficient-memory.

The last question:

How can I free that space?

• If the space belongs to Internal SD card, then you have multiple options to delete the unnecessary stuff. You can either use DiskUsage which also gives Delete option under menu, or delete the content using your preferred File Manager, or use PC to delete the stuff.
• If the space belongs to /data then you'll have to uninstall the apps.