We all know that objects at a lower orbit have a higher speed and objects at a higher orbit have lower speed.
If you're a spacecraft trying to get from a lower orbit to a higher orbit, you would accelerate two times.  
Once at your periapsis, burning prograde.  
Then once at apoapsis, burning prograde, to raise your periapsis.

Any school student can tell you doing two accelerations in your direction of travel will obtain you a higher linear speed.
So how is this possible?

I did some math. [Say you're at Kerbin](https://wiki.kerbalspaceprogram.com/wiki/Kerbin) with an 85km circular orbit. You want to get to a 100km circular orbit.  Orbital speed at 85km is 2270 m/s. Orbital speed at 100km is 2246 m/s. 
By the vis-viva equation, speed at periapsis is 2283 m/s, speed at apoapsis is 2234 m/s.  1st burn: 2283 - 2270 = +13 m/s.  2nd burn: 2246 - 2234 = +12 m/s.  How do you explain what's going on?