This is probably a bit confusing for Americans because in the UK big cabinet posts in that executive will be filled by people from the legislature. So all the senior jobs go to elected representatives - but the citizens who elected them don't get to pick who gets which job.
Imagine if Mitch McConnell runs the executive, he gets to pick Senators for important stuff like the Secretary of State, and lesser things like the top of the FCC go to Republicans in the House. They're still in Congress, they just are now also running an executive. Just as now all the real work is done by career civil servants, the politicians are there to set political direction.
The Palace where the UK's elected legislators meet is divided down the middle so that members of the political parties that aren't in Government are quite literally "the opposition" - they're facing the members from the governing party.
Because power might shift suddenly a serious opposition political party has to have "shadows" of every major executive role. A Shadow Environment Minister for example doesn't have a huge Department actually governing the environment like the "real" Environment Minister but they need to know what's going on in that department and be able to explain how their party would do it differently, because tomorrow they could be in power.