The "kh" is not meant to be understood as a sequence of "k" followed by "h". The idea is that it is a digraph where both letters together represent the single sound of Russian х. It is used by analogy with the English digraphs "th" and "ph", which also are used to represent single fricative sounds. (Those digraphs are based in part on old traditions about how Greek is transcribed.)
English "h" by itself is not pronounced exactly the same as Russian х. The English sound is categorized as a glottal fricative (written [h] in the International Phonetic Alphabet) while the Russian sound is categorized as a velar fricative (written [x] in the International Phonetic Alphabet). Most English speakers aren't accustomed to pronouncing the velar fricative. The glottal fricative /h/ is probably the English consonant that sounds the closest to the velar fricative [x], but some speakers might use the velar stop /k/ as a replacement instead, either because of influence from the spelling or because the velar fricative sounds like [k] to them.
Going the other direction, I've heard that it is usual for Russian speakers to approximate English [h] with Russian х, but it is an approximation, not an exactly identical sound.