Can one know the veracity of a claim if they were not there to experience it first hand

It is important to understand the difference between an experience and a claim, and their relationship to one another. We experience sensory data that requires interpretation. Once interpreted, we translate our experience into words that makes the claim. The human brain doesn't make it easy for us to do this, thanks to cognitive biases. This means that multiple first-hand witnesses to an event (i.e., those who experience it) can make contradicting claims about the event. We also need to consider purposeful deception (lying). The bottom line, first-hand accounts are generally better then hearsay in terms of determining veracity, but this is only one factor to consider.

Investigation using the scientific method or rigid journalistic standards streamline the process of deriving a claim from an experience or observation, but also can test the veracity of a claim by evaluating the evidence for the claim. It is the process of properly evaluating evidence that we verify first hand accounts and personal experiences.

This is a rational explanation as to why what they claim is false.

There is also a logical explanation as to why what they claim is false. Your mutual friend made a self-refuting statement or made a statement with conflicting conditions .

nobody knows what's true or false unless they were there first hand and experienced it.

According to the claim itself, your friend would have to have been "there" to experience "nobody knows what's true or false unless they were there first hand and experienced it." This is an oddly meta and perhaps impossible claim. This is more of a "gotcha" response, but I believe the rational response is much more helpful in terms of helping your friend improve his or her critical thinking.

Yes. 

Knowledge can come from:

a) sensory experience ("first-hand" knowledge)

b) communication ("second-hand knowledge", think news media)

c) reason ("intuition")

It is wrong, therefore, to assert that one cannot determine the veracity of a claim if they did not experience it first hand, because personal experience is not the only form of gaining knowledge.

As Dr Bo points out, this is also contradictio in adjecto as "nobody knows what's true or false unless they were first hand and experienced it" is asserted as true without first hand experience of the statement in action, rendering it self-contradictory. The same problem arises when people try to claim that "there is no objective truth", as the statement itself is asserted as objective when this would blatantly clash.