I don't understand how around a fifth of Israeli citizens are Arabs or "Palestinian citizens of Israel" if Israel kicked all the Palestinians out when it gained its independence.

Simply put, they didn't. While the vast majority of the non-Jewish Palestinians fled or were expelled (and I don't want to get into the hornets of nest of that debate) from the newly formed state of Israel, a substantial portion did not not. Cities like Jaffa (today, de facto the Arab part of Tel Aviv) came under Israeli control but remain Arab. The Negev Bedouin also form a substantial part of the Arab-Israeli population. The largest proportion come from the north of the country, as you can see in this map here, in areas that were taken under Israeli control in 1948. Another sizable proportion are those Arabs that were offered and accepted citizenship after 1968, particularly in the Golan and Jerusalem, though many did not accept the offer.

Anyways, in the broader Arab-Israeli or Israeli-Palestinian conflict this population is often brought up in comparison with not only the overall Jewish population but also with the Jewish but non-Haredi population, and then compared with the population of the West Bank and Gaza, particularly when considering the various one or two-state options.