Here's an ELI5:

The Constitution

It is true that under Article 1, Section 8 of the Constitutional, only Congress has the power to declare war. But a singular strike against a limited asset hardly constitutes war. It is a military action intended to confront hostilities which does not rise to the level of a declared war as the Founders envisioned.

It is clear that the Framers recognized there would be times when the President needs to use immediate military force unilaterally and without the extensive time it might take for Congress to consider the matter, then debate it and approve it. They chose to withhold from him only the power to declare war, not make war which was regarded as a vital emergency power allowed the president to counter or thwart foreign threats.

Hence it is perfectly consistent with the Constitution for President Trump to have taken action against the brutality of Bashar Al-Assad to prevent the spread and repeated use of chemical weapons. It is not only Syrian civilians the Commander-In-Chief is endeavoring to protect, but U.S. troops in the region and other American assets, all of whom stand in jeopardy.

Congressional Acts

In 1973, Congress passed The War Powers Act in reaction to the Vietnam War, which was never a declared war even though roughly 60,000 American soldiers were killed.

In relevant part, the law permits the president to launch a military strike on his own, as long as he notifies or consults with Congress within 48 hours. This President Trump did.

An argument can also be made that the president has latitude to take action against Syria under a 2001 law passed by Congress following the 9/11 attacks. Called The Authorization for Use of Military Force, it gives him permission to direct military force to prevent future terror attacks against the U.S.

Presidents Bush, Obama and Trump have used an expansive view of this authority to fight both Al Qaeda and ISIS in Afghanistan, Iraq, Syria and elsewhere.

The law is not limited to terror groups. It encompasses any nation which engages directly or indirectly in terrorist acts or harbors terrorists.

The Syrian chemical attack against innocent people can surely be described as an act of terrorism. And again, the U.S. has vulnerable troops in the region helping in the fight against ISIS. They might be the next target of an Assad chemical attack should he not be deterred by the use of American force.