Economic engagement basically means economic interactions with another country that are meant to try to help reform that country.
Mexico today has many problems. The most significant of these problems, from the point of view of the United States in particular, is the terrible violence that is being fuelled by drug smuggling cartels. The US government would like to reduce this violence, particularly since there is some danger of it spilling over into the US. The question is how to do this. One possible way is through economic engagement.
When we talk about “engagement” with another country, we are talking about trying to work with it rather than abandoning it. Economic engagement involves working with the other country through trade and other such interactions. The hope is that if we have good economic interactions with Mexico, we can gradually help them deal with their problems.