The attachment manager allows you to attach fonts and/or pictures to your script (by encoding them as text). This is occasionally useful for sharing fonts between everyone who is working on a script without having to send the fonts as separate files. However, it is quite limited and is prone to causing problems.
The entire dialogue is fairly self-explanatory. The two "Attach ..." buttons add attachments, "Extract" extracts existing attachments into separate files, and "Delete" deletes attachments from the subtitles file.
The SSA format specification only allows certain filetypes to be attached. For fonts, only .ttf is allowed. For pictures, .bmp, .gif, .ico, .jpg and .wmf are allowed (note the absence of .png). None of the SSA commands which use the images are implemented in anything but SubStation Alpha, so it is very unlikely that attaching pictures is actually a useful thing to do.
Many SSA/ASS editors ignore or strip attachments. The original SubStation Alpha (v4.08) will work fine, but only for real SSA files. Sabbu will complain about unrecognized fields, and strip the attachments if you save the file. Most other editors either ignore the attachments or crash when encountering them.
A notable exception is mkvmerge, which will convert the attached files to Matroska attachments on muxing. If you demux the script again, the attachments will be stripped from the script, but they're still there as MKV attachments.
Unfortunately, storing binary data as text (in this case, a variant of UUEncoding) is not very efficient. The attached files will take considerably more space as script attachments than they do as separate files.
Aegisub's undo system makes a complete copy of the loaded file on every change. Normally this is very fast, but attachments can significantly slow this down due to the large size.