Why am I getting 24 volts in kitchen outlets, unless TWO breakers are on?
This is a multi-wire branch circuit

aka MWBC.  This is a special arrangement for delivering two circuits' worth of power onto a single cable with only 1 more copper wire, because it is putting two opposite-pole subcircuits arranged so the neutral only carries differential current.  It's quite wire-efficient, but yours was installed quite improperly.  They pretty much did everything wrong.  

Also the six GFCI receptacles were not original; they were added probably by a previous seller because the home inspector red-flagged it.  If you know anything about GFCIs, you only need 1 per circuit, but an MWBC is one circuit -- and the way they share neutral requires either one GFCI at the breaker, or a GFCI at each receptacle (expensive). 

The first problem is that MWBC requires a handle-tied or 2-pole breaker.  You don't have one.  In fact, you haven't said, but you might even have the worst nightmare of MWBCs, which is a duplex breaker. 

Fix the breaker... and revisit GFCI

OK, so the breaker needs to be replaced with a 2-pole breaker.  (you can do handle-ties, but those are a PITA to find, and a 2-pole breaker is available pretty much anywhere.) 

However, remember where I said an MWBC can have either 


one GFCI at the breaker... or 
six GFCIs at every receptacle separately? 


When you have more than 4 receps, it's actually cheaper to fit a single GFCI+breaker.  Around $20 each for six GFCIs vs about $80 for a 2-pole GFCI breaker ($90 if you want AFCI also). 

The only downside of this is that a GFCI trip with receptacles will only trip the single recep (and you can reset it right there), but a GFCI trip with a breaker will knock out everything, which may be confusing and will send you on a trip to the basement to reset.  Your call. 

But you need a 2-pole breaker.  Remember, breaker types must match the panel.  If the panel is Murray QP but the breaker is Eaton BR, the breaker is wrong and replace with a Siemens/Murray QP not a BR.  Alien breakers are dangerous and burn up panel buses.  This is especially important when spending $80 on a breaker!

If you currently have a duplex breaker (two handles in 1 space) then ask a new question about how to replace that with a 2-pole.  

Neutrals need to be pigtailed

You didn't mention this, but the neutral wires in the boxes need to be spliced with a wire nut etc. and a single neutral wire brought out to the receptacle.  

Removing a device must not sever power to the other half of the circuit. 

However, it's perfectly OK to use the GFCI receptacle as a splice point for the hot wires.  Note that ALL these wires will go on the LINE screws (most accept 2 wires in back-wiring), the LOAD terminals will never be used here.