1. It is not clear that they are illegal. Legal scholars disagree on whether they are legal or illegal.
2. They are building primarily on land that is either private property owned by Jews or government owned land. The land was ethnically cleansed of Jews in 1948. (Depending on who you ask, the land is either 100% or 80% Jewish Private property and/or govt owned property.)
3. So, the land does not "belong" to someone else. Rather, the Palestinian Arabs want that land to become part of a future Palestinian state. Under UN SC Res 242, that land does not "belong" to any particular state (neither Israel nor Palestine nor Jordan, nor anyone else). The whole point of UN SC Res 242 is that the land is in legal limbo (sovereignty-wise) and Israel may administer it until and unless all relevant parties peacefully negotiate final status borders.
4. Israelis (and many others) don't believe that the settlement building really is a key issue.
Their Arab neighbors were trying to destroy Israel in 1967, which is how Israel ended up with it.
Israel immediately offered to trade it all for peace. No takers. Instead, the famous "three no's of Khartoum."
The PLO was formed and began terrorist attacks on Israeli civilians in 1964, 4 years before Israel had the WB or Gaza and so no settlements.
They tried to destroy Israel in 1948 (which is when they ethnically cleansed the Jews from Jerusalem and the West Bank and Gaza.
They ethnically cleansed all the Jews from 80% of Palestine in 1946, when it was granted independence and renamed Jordan.
They ethnically cleansed the Jews from Hebron in 1929 (about 20 years before Israel...)
And they were slaughtering Jews throughout the 1920's...

