The laryngeal consonants were guttural sounds produced in the back of the throat, like English h, German ch, Arabic `ayn or qaf, etc. We don't know exactly what the pronunciations were, but in other existing languages, laryngeal sounds like this tend to have an effect on the surrounding vowels. This happens because your tongue has to movie continuously between the sounds it produces, and so there's some blurring that happens between adjacent sounds.

In particular, laryngeal sounds tend to make vowels near them become more back, low, and round. So for example before a uvular q sound in Arabic, the sound /a/ which is usually something like English a in cat will become back and low, something more like au in caught. You can get a process like this, playing out over hundreds of years:

1. You start with a form like Hewis, where H is some laryngeal sound.

2. The laryngeal sound makes the following vowel low and back, yielding Hawis.

3. The laryngeal sound drops, leaving just awis.

In fact this is how we get the Indo-European root for "sheep" (reflected in English ewe, Latin ovis, etc.). The Hittite form is hawis.

See the video linked in the sister comment for (IIRC) an excellent discussion of historical reconstruction of sounds from writing systems, but I can add some clarification to the matter of laryngeal consonants: they are confusingly named for historical reasons but consitute three unknown consonants with a place of articulation towards the back of the vocal tract (near the larynx). In addition to this reconstruction, this specific case has the additional matter of finding consonants in a place where they are predicted from their influence on the nearby vowels in IE languages (and, incidentally, the expected vowels), but where there is otherwise no consonant attested in non-Anatolian IE languages.

Laryngeal refers to the place of articulation (where the air stream is constricted). They are pronounced way in the back of the throat, like the glottal stop. Even further back than things like Greek chi or German ch.

For an interesting example using ancient Egyptian, see https://youtu.be/J-K5OjAkiEA