Frankly, I think the answer is that many, if not most, people would simply disagree with the assertion that state and local elections are more influential than federal ones or that Congressional elections are more influential than the Presidential ones.
Federal vs. State/Local
In particular, state or local laws cannot be enforced if they would violate federal ones, so, if a matter is decided at the federal level, then your state and local politicians are powerless to do anything about it (other than, of course, voting and encouraging people to vote in the federal elections.) In particular, Article VI, paragraph 2 of the U.S. Constitution (commonly known as the Supremacy Clause) states that:
This Constitution, and the laws of the United States which shall be made in pursuance thereof; and all treaties made, or which shall be made, under the authority of the United States, shall be the supreme law of the land; and the judges in every state shall be bound thereby, anything in the Constitution or laws of any State to the contrary notwithstanding.
Presidential vs. Congressional Elections
As for why people tend to be more likely to turn out to vote during a Presidential election than in a mid-term election (i.e. a federal congressional election during years where the Presidency is not on the ballot,) there are a few reasons for this.
Power of the Presidency
Power Over New Legislation
One reason is that the Presidency concentrates far more power into one individual than any other position at any level of government in the United States. While the President cannot technically make laws on their own, their assent is required for any new laws to be made by Congress unless 2/3 of each house of Congress votes to override the President's veto. In practice, that almost never happens, so the President's consent is required for virtually any legislation to pass. That is, even if you can get just under two out of three members of each house of Congress to agree on a law, if the President does not agree, Congress cannot pass that law.
Additionally, there are many existing federal laws which grant a large degree of discretion to executive branch agencies. Within the (often very wide) bounds of those laws, the President can unilaterally issue Executive Orders. This includes things all the way up to deploying the U.S. military for up to 60 days under the War Powers Act, for example. But it also includes the power to shape the absolutely vast Code of Federal Regulations within the bounds of the laws authorizing regulatory agencies. According to the Mercatus Center at George Mason University, as of 2013, there were 235 entire books of federal regulations with an average length of approximately 750 pages each. They put together this interesting infographic comparing the word count in the Code of Federal Regulations in 2013 to various (entire series of) books:
Code of Federal Regulations - Image Source: Mercatus Center, George Mason University
At 103 million words as GMU's work in 2013, the CFR was approximately 218 times as long as the entire Lord of the Rings trilogy or 130 times as long as the Bible. (And the imagery isn't nearly as good as Tolkien's...)
If the President decides to direct federal agencies in a particular manner, then, as long as its within the bounds of the laws that create the authority of those agencies, there's nothing even your U.S. Representative or Senator can do to stop them unless they have 2/3 of both Representatives and Senators willing to pass a new law.
Power of Appointments
The President has the sole power to nominate the entire Cabinet (i.e. heads of the various executive branch departments) as well as all federal judges. These and a few other positions (e.g. members of the Federal Reserve Board) must be confirmed by the Senate in order to take office, but the President is the only one who can nominate them and the Senate can only agree or disagree to the appointment. Additionally, there are lots of less-important executive branch positions that the President can just appoint directly without any need for confirmation by the Senate.
So, while the House and Senate do indeed have considerable power, an individual member of the House or Senate doesn't wield nearly the power of the President.
Uncompetitive Congressional Elections
Additionally, many House and Senate races aren't nearly as competitive as the race for President.
Thanks in part to major natural regional differences in prevailing political views (e.g. inner city vs. suburban vs. rural) and in part to gerrymandering, many House seats really aren't competitive in the general election at all. For example, in my district, the candidate from the more popular party here has received more than 2/3 of the votes in the last 6 consecutive elections (i.e. since 2010.) When the winner of a race is a complete foregone conclusion, people won't be as motivated to vote.
While Senate seats can't be gerrymandered (because they are statewide elections,) many of those races still end up being quite uncompetitive due to many, if not most, states having a very significant bias in favor of one major party or the other.
While there are some House and Senate elections that are extremely competitive, overall turnout in mid-terms tends to be reduced by lower turnout in the districts without a competitive race on the ballot.