Best way I've seen it explained.

FLAC = PNG

MP3 = JPEG

FLAC files are typically larger than MP3, because with MP3 roughly 90% of the original file is discarded on encoding, hence lossy compression. At decoding the codec tries to restore this 90% out of the original remaining 10%. This sounds worse than it actually is, but you understand why MP3 is considered inferior by audio purists.

On the other hand FLAC uses a different compression technique that reduces the original file by 30-50% but without any data deletion, hence lossless. So, original file let's say 100MB as .wav, compresses to 30-50 MB IN FLAC and 6-10 MB in MP3.

I gladly sacrifice the additional storage to get noticeably better quality audio.

In term of data retention, FLAC is like compressing a RAW image file by using a zip/rar, the data integrity is the same when you open the zip/rar and check the image. MP3 is like reducing the size of a RAW image by converting it to JPEG, data integrity is affected and the image won't look as good anymore. The most common uncompressed audio format that people know is WAV files (but you can also find compressed WAV files).

FLAC files use lossless compression, which means all the original data is preserved (which is why people like FLAC), which makes FLAC files smaller than an uncompressed .WAV audio file, but still larger than an MP3 file, which uses lossy compression. Lossy compression does involve the loss of some data. This is what allows MP3 files to be smaller than FLAC. With the existence of other more modern lossy audio file formats/codecs like .AAC, and .opus, MP3s should no longer be used unless it is required for compatibility reasons. The modern lossy formats offer higher quality audio files at lower file sizes than MP3.

Yes. Mp3 is not lossless, so it can be even smaller

MP3s are quite compressed, meaning a lot of data is thrown away in an effort to have smaller files. The quality of audio is sacrificed quite a bit though.

Lossless formats retain as much data as possible as to not impact the sound quality, but at the expense of larger files. The OP says "smaller" because that's in comparison to the raw uncompressed sound data stream. But they are larger than MP3s because MP3 is a lossy format.

File size used to matter a lot in the past when digital music players first came out. My first player had 128 MB storage, for example. At 3-5 MB per song that would fill up quickly. Nowadays larger storage of portable devices is more ubiquitous, with even the cheapest phones sportiing 32-64 GB, and more. So people prefer audio quality and don't care as much if each song takes up more space.