The major strength of the legislative branch is that it is the branch that actually has the right and the power to make laws. In the national government, this is the Congress. The Constitution gives Congress the right to make the laws. No other branch of government can actually make laws. Congress can make any law that it wants on any topic that it wants (with a few exceptions listed in the Constitution). This gives Congress a great deal of power.
However, Congress does not really have the power to lead. Only Congress can make the laws, but Congress is very weak when it comes to trying to encourage the people to support a given agenda. If the leadership of Congress wants to push a certain agenda (imagine, for example, that it wants to pass a health care program to replace Obamacare), it has a very hard time leading the public and getting the public to support the idea. This is because Congress is so big and has so many members. No individual member tends to have much in the way of name recognition with the public. No individual member of Congress leads Congress and can speak for the whole body. This means that it is very hard for Congress to get out and lead people to support a particular policy.
Thus, Congress is strong because it has the power to actually make the laws, but it is weak in that it cannot really lead the people and get them to support a particular agenda.