Egypt has taken initiative to hold peace talks between Israel and Palestine on various occasions. What is their interest in the matter? Is it simply a case of proximity, or does conflict between Israel and Palestine cause internal conflict within Egypt? Or, even more complicated, is Egypt being utilized by Western powers?
Being peace maker enhances Egypts' position.
Ever since Gamal Abdel Nasser, Egypt was interested in being the leader of Arab world.
Playing peace-keeper in what's considered as the among the most consequential conflict grants both actual (everyone owes them favors) and perceived (they are a leader because they are able to do this) power.
Egypt is definitely affected by any conflict between Israel and HAMAS (which is an offshoot of Muslim Brotherhood Egyptian organization - google how they relate to the last 5 years in Egyptian politics)
They are definitely hoping that Western powers will be more favorable towards them - see the "favors accrued" point in #1 bullet. That's not a push from Western government per se but that is a factor in their calculations.
Any disruption in Gaza strip negatively reflects on Egypt (terrorists going to Egypt; refugees escaping there, etc...).