How do historians verify historical claims?

Just like Scientific method exists to prove or disprove scientific theories and hypothesis, Historical Method also exists.

Historians of course cross-check certain claims with contemporary sources including archaeological evidence and thus proceed to create their account of the concerned historic event.

Source Criticism

First step to this is called Source Criticism. According to "A guide to Historic Method" by Garraghan, following aspects are checked in this step:

1. When was the source, written or unwritten, produced (date)?
2. Where was it produced (localization)?
3. By whom was it produced (authorship)?
4. From what pre-existing material was it produced (analysis)?
5. In what original form was it produced (integrity)?
6. What is the evidential value of its contents (credibility)?

The first four are known as higher criticism; the fifth, lower criticism; and, together, external criticism. The sixth and final inquiry about a source is called internal criticism. Together, this inquiry is known as source criticism.

Bernheim has however proposed seven inquiries for this step which includes search for contradictory sources.

R.J. Shaffer proposes to take Eye-Witnesses into account as well. Garraghan further expands on it to include indirect-witnesses & oral tradition.

Synthesis: Historical Reasoning

Second step is called Historical Reasoning which involves drawing best possible logical conclusion from results of Source Criticism.

There are three aspects to it:

1. Argument to Best Explanation
2. Statistical Inference
3. Argument from Analogy

C. Behan McCullagh lays down seven conditions for a successful argument to the best explanation:

1. The statement, together with other statements already held to be true, must imply yet other statements describing present, observable data. (We will henceforth call the first statement 'the hypothesis', and the statements describing observable data, 'observation statements'.)

2. The hypothesis must be of greater explanatory scope than any other incompatible hypothesis about the same subject; that is, it must imply a greater variety of observation statements.

3. The hypothesis must be of greater explanatory power than any other incompatible hypothesis about the same subject; that is, it must make the observation statements it implies more probable than any other.

4. The hypothesis must be more plausible than any other incompatible hypothesis about the same subject; that is, it must be implied to some degree by a greater variety of accepted truths than any other, and be implied more strongly than any other; and its probable negation must be implied by fewer beliefs, and implied less strongly than any other.

5. The hypothesis must be less ad hoc than any other incompatible hypothesis about the same subject; that is, it must include fewer new suppositions about the past which are not already implied to some extent by existing beliefs.

6. It must be disconfirmed by fewer accepted beliefs than any other incompatible hypothesis about the same subject; that is, when conjoined with accepted truths it must imply fewer observation statements and other statements which are believed to be false.

7. It must exceed other incompatible hypotheses about the same subject by so much, in characteristics 2 to 6, that there is little chance of an incompatible hypothesis, after further investigation, soon exceeding it in these respects.

McCullagh states Statistical Inference as follows:

1. There is probability (of the degree p1) that whatever is an A is a B.

2. It is probable (to the degree p2) that this is an A.

3. Therefore, (relative to these premises) it is probable (to the degree p1 × p2) that this is a B.

The structure of Argument by Analogy is as follows:

1. One thing (object, event, or state of affairs) has properties p1 . . . pn and pn + 1.

2. Another thing has properties p1 . . . pn.

3. So the latter has property pn + 1.

The book by Garraghan will help you a lot to understand the process. Other than that, this brief PDF paper and its sources will also come handy.