The question asks why some people think President Barrack Obama shouldn't pick the next US Supreme Court Justice. The short, overview answer is that they believe that anyone nominated by the current president is likely to be of a more liberal political persuasion, and as such would tilt the balance of opinion on the court in favor of a liberal interpretation of the US constitution. To the extent that this is viewed as unfavorable by people, they would like to see the choice of justice put off until the next administration is installed after the elections occurring in November of this year. In essence, this constitutes a political gamble, namely that the next president will be more conservative politically than President Obama. It also rests on the assumption that it is possible to predict the political direction of a Justice’s opinions.
On this last point, it is important to note that Supreme Court Justices interpret the constitution and, as such, they are expected to avoid becoming a de-facto unelected legislative branch. The debate about specific candidates then often revolves around the extent to which they feel bound by what they believe was the original intent of the framers of the various elements of the Constitution (including its Amendments), versus the view that it is a “living, evolving document”. Another debate often revolves about what the candidate believes regarding the limitations on the role of government; modern conservatives generally favor greater limitations on Federal power, liberals favoring more expansive Federal roles.
Finally, in managing public opinion, protagonists on both sides of the issue have attempted to posit various precedents regarding the handling of nominations by previous presidents nearing the end of their terms. Legally, none of these past treatments are binding in any way on current administrations or legislatures.