Well the Constitution vests the entire power of the Executive into the President and Vice-President, and they two alone run it.
Now of course, not even the people who founded the US thought two people could run the entire country, which is where Article 2 Section 2 comes into play. However, there is a stipulation. The President, if he or she wants to "delegate" a part of their executive power, they must do so with consent of Congress. The President just can't say, "Oh this person is going to be responsible for X, Y, and Z." Congress has to approve that handing of power over to someone.
Now I'm going to skip a whole lot of other stuff in the Executive branch like commissions and groups and "defer" actions and what-not. But those exist as well and have other rules and are established via a whole slew of different means. They all play vital roles in the grand scheme of things and you should remember that me skipping all of that, makes this a gross simplification. I know hate to simplify, but it's a lot and I'll just have to take the hits I'm sure I'll get like "Oh you forgot!". So just going to apologize for such a bastard version ahead of time here.
So anyway, these departments, the President gets to say when he needs one and Congress whips up a law establishing the official transfer of that specific power to such and such department (at least that's the extremely simplified version of that). Each department has a senior officer who is appointed by the President and confirmed by the Senate, that's the Cabinet member (there also a whole lot of other ones within each department that have to get confirmed, I'm skipping them but they exist as well). That guy or gal gets to call the shots in the department on how that department is to carry out their function. The people the senior staff hires are Federal employees and do not require confirmation. Federal employees can be fired by the senior staff. However, Congress has passed laws about Federal employees and set up rules for when they can and cannot be fired.
Without getting too detailed, it boils down to you can only fire a Federal employee for "good reason". Now some would cite unions elbowing their grubby noses into the mix for that, some would cite holy whistleblowers and how they saved us for that, the real story is that it's a mix of all of the above really and makes for great history classes. Now what is "good reason" in some cases is pretty clear, you can't fire someone for exposing theft or corruption, in a lot of other cases it's less clear and requires a judge to step in, cue the Judicial branch. What Trump wants to do is remove some protections that Congress has given Federal employees. Now no one knows which specific rules Trump will target but it was made clear that he wants to "open up" the firing process to be a bit more liberal than it currently is. And based on the flavor of Congress/President mix that you have, those rules have changed a lot. Typically when Congress and the President are the same flavor you get the most changing in rules.
What I think is interesting (some might say horrifying) is the proposal to fire people whose ideology doesn't match with the current administration. Now Trump hasn't said that exactly, but handing out questionnaires like the one talked about isn't building a lot of confidence. Typically, a department wants to rely on their staff to make informed decisions (which in theory is why some higher up employees in the Federal government but not the senior staff can go tell the President to get bent [key words here "in theory"]), have those bubble up to the senior staff, and then have the senior staff pass those on to either rule making or the Presidential consideration. If someone is in the department who doesn't agree with the majority ideology, typically their ideas just don't get "bubbled" upwards. Firing people because their ideology doesn't match up is just setting a department up for revolving door syndrome. Additionally, not having good professionals in your department typically means that the business sector will overtake the government on leading the way. This was long a criticism for technology and (personal opinion here, insert grain of salt) I still think the Government is still behind the ball on technology. So if you get rid of the real professionals from your department, it doesn't mean that suddenly the Government will get it's way, it just means that the private sector will soon be calling the shots instead of the public sector.
Anyway, I find it all amusing. If anything it'll be interesting wordmsithing to be soon had.