The rules don't say that player characters need to kill enemies to earn XP.
From the Core Rulebook, chapter 12 "Gamemastering" gives an overview of how to run combats and combat rewards. The text says that the players must "defeat" monsters or "overcome" the challenge in some way, but does not explicitly say that the enemy creatures must be killed by the players, let alone killed by anyone. On page 399, the section "Awarding Experience" says:
Pathfinder Roleplaying Game characters advance in level by defeating monsters, overcoming challenges, and completing adventures—in so doing, they earn experience points (XP for short). [...] Each monster, trap, and obstacle awards a set amount of XP, as determined by its CR, regardless of the level of the party in relation to the challenge, although you should never bother awarding XP for challenges that have a CR of 10 or more lower than the APL.
According to James Jacobs, one of the system's designers, enduring an encounter counts as the first time the players overcome it, regardless of whether the enemies survived. From the Paizo forums:
ALL XP awards in the game are handed out when the thing they're attached to is defeated. Be that a monster or a trap or a haunt or a tense political standoff. Be "defeated" akin to "killed" or "driven away" or "disabled" or "endured.
There are various gameplay consequences due to your GM's rule that XP is only given for killing blows. Players wouldn't gain XP if they rely on summoned creatures, mind-affecting spells (such as Confusion or Dominate Monster), or environmental objects to dispatch enemies. This rule doesn't account for non-creature hazards such as haunts or traps, which have CR values, and are common in Rise of the Runelords. It also discourages diplomacy, stealth, and other non-combat methods of handling encounters.
However, the GM decides the rules for rewarding XP.
The Gamemastery Guide, chapter 5 "Rewards" has some rough rules for how and when XP should be given to the players. The section is quite long (page 102-104) and begins with the following:
Experience points are the lifeblood of the Pathfinder rewards system. They determine the rate at which the PCs progress, and form the currency with which the most spectacular and reliable abilities are acquired. By deciding when and how to give out XP, you’re establishing the expectations the players will bring to the rest of the game’s reward system.
To summarize, the gist of the section is that the GM judges what encounters and conditions are worth rewarding XP. So if the GM rules that the players only get XP by delivering the "killing blow", then it's legal, even if the rulebooks don't support it.
What can you do about this?
As usual, the solution is "Talk to the GM." The GM is treating combat like a video game, and that's not working for you. Explain that you don't like this rule about killing blows. Not only because it's unsupported by the official rulebooks, but more importantly, that you think it's unfair to be withheld XP even after your characters endured a challenge or encounter.
Side note: Rise of the Runelords is an especially tough adventure path, with numerous encounters where an enemy NPC may flee. Falling behind on XP (and other resources) can make future encounters much more deadly. The players will need as much XP as they can earn.
When you talk to the GM, it may help to get input from the other players. Come up with a new agreement for how and when XP is rewarded.