The energy for a jet engine comes from burning fuel in the combustion section. That energy does two major things: the turbine taps some of it to spin the engine (which includes the compressor, the accessory drive, the N1 fan, and, on a turboprop the really big & unducted fan out front called the prop), and the rest exits the back as thrust.
The more work the compressor is being called upon to do, the more fuel you have to burn to generate the energy to do that (assuming that you are keeping the thrust part of the output constant). Burning more fuel = higher EGT.
When you tap, or bleed off, some of that compressed air for things in the pneumatic system to use, that increases the work that the compressor has to do, compared with a baseline case where it compresses enough air for the combustion section but only that.
The maximum power available, the max N1 you're allowed, is greater when you aren't drawing that bleed air off -- and so you see the N1 limit value on the CDU go up when you turn the packs off. But that is the limit. At that higher limit, with the packs/bleeds off, your engines are producing more thrust than they are at the lower limit that's shown when the packs & bleeds are on.
If you need a certain amount of thrust to meet your requirements (accelerate/stop, accelerate/go, second segment climb, etc), then you could get that level of thrust at a similar or slightly lower N1 with the bleeds off than with the bleeds on. But, generally, flight planning doesn't work that way; if a bleeds-on takeoff gives you what you need, then you do that (and may also use less than full thrust -- with assumed temperature or derate or both). In the case where even full bleeds-on doesn't give you enough power, then you get a bleeds-off solution, and those generally don't use reduced thrust.
It's not impossible to have a reduced thrust bleeds-off takeoff, but it takes a performance calculator that gives you pretty granular control of the parameters, and there's pretty much no need to ever have both of those at once. And so the pilot-proof versions of things generally assume that if you're requesting bleeds off data, it's because you want all the performance possible, and so they give you the full power solution. But, if you had a way to specify "this much" power & no more, then you'd see that the bleeds-off N1 to give you just that much power would be either the same or a lower N1 (and certainly a lower resulting EGT) than the bleeds-on solution.
