Thinking critically about a claim involves interpreting it correctly, accepting or rejecting it only for good reasons, and drawing reasonable conclusions from it.
A critical thinker has an attitude--an attitude of desiring to avoid nonsense, to find the truth and to discover the best action. It's an attitude that rejects "intuiting" the truth in favor of demanding reasons. To be a critical thinker you need to be fair and open-minded even with people you disagree with. You need to give them a fair hearing because your goal is the truth or the best action. Your goal isn't just to confirm what you already believe.
Your adopting the attitude of the critical thinker is essential if you are to avoid the garbage on the path of life.
But a critical thinker is not a hypercritical thinker. The point isn't to be so critical that you find fault where there is no fault or that you make mountains out of molehills by overstating small problems.
Also, if you are a critical thinker, you know how well you know. That is, you know how strongly you should believe. For example, you know math, yes, but only some math, not all math. Your knowledge might qualify you as an expert on fifth grade math homework, but not on deciding which articles to publish in a professional mathematics journal. In other words, you have a sense of the limits and qualifications on your knowledge.
The following is a brief, but excellent, definition of "critical thinking" from a bill in the California State Senate that was trying to update the State's Education code: