1. h₁, h₂, h₃ represent laryngeal (guttural) consonants. When in the beginning of the 20th century linguists discovered Hittite language, it turned out to have more guttural fricatives than just [h], like in most of known IE languages. It is now believed that PIE had a few different kinds of guttural fricatives, but since linguists don't know for sure what exact phonetic sounds those were, they just mark them with numbers. You can easily find more information in the subsection on laryngeals in PIE in en.wiki.
2. k̂, ĝ, ĝʰ represent palatalized consonants. In PIE, they probably were just allophones (positional varieties) of k, g, gʱ, not distinct phonemes, but since they often have separate reflexes in a lot of descendant-languages, linguists decided that it's important enough to mark them by separate characters.
All this information can be easily found in en.wiki or elsewhere in internet. Or you can ask people in /r/linguistics for more elaborated explanations.