The good direction
Most format conversions are a compromise. This one mostly is not. A WAV holds uncompressed PCM, meaning every sample exactly as it was captured, with no codec having made any decisions about it yet. So to convert WAV to MP3 is to run a first-generation lossy encode, and lossy codecs are at their best when they can see the real waveform instead of the leftovers of somebody else's compression.
That is why the advice here is different from the advice on an M4A or OGG page. There, re-encoding stacks a second round of loss on top of a first. Here, there is only one round, and at a sensible bitrate the result is genuinely hard to distinguish from the source on normal playback equipment.
What you are trading
Size, for detail you probably cannot hear. Uncompressed stereo at 44.1 kHz and 16 bits costs about 10 MB per minute. The same minute at 320 kbps costs about 2.4 MB, and at 128 kbps about 1 MB. Nothing about the audio gets better in exchange, so the only question is how much you are willing to give up.
The parts of a mix that reveal MP3 first are the ones with the least structure for the encoder to model: cymbal decays, room reverb, crowd noise, breaths between words. Dense, loud material hides encoding artifacts well. Sparse, quiet material does not.
When to keep the WAV
Keep it if the file is a master, if it is going into a mix or an edit, or if anything downstream will re-encode it again. Every lossy pass compounds, and the right shape for an archive is a lossless original with lossy copies made from it as needed.
Files are read and encoded by your own browser. The engine is ffmpeg compiled to WebAssembly, downloaded once and then cached, so nothing about your audio travels over the network.