to press physically: to squeeze or crowd something so that it is pinched tight
Underneath every use of this word is the feel of something bearing down. Greek writers reached for it when armor chafed a soldier, when a beggar rubbed his shoulders past a doorway, and for the spot where a shoe pinches. Two places in the New Testament keep that plain bodily sense.
In Mark the crowd around Jesus grows so thick that the disciples keep a boat ready so he will not be squeezed by the press of people. In Matthew it is not a person but a road: the way that leads to life has been pinched narrow, and the sentence sets this word beside another one meaning "narrow" (στενός) so that the gate and the road reinforce each other.
Greek could sharpen the picture by building on this verb. One such word describes a crowd squeezing in from every side (συνθλίβω); another describes the hard shoving Peter complains about when Jesus asks who touched him (ἀποθλίβω). This is the plain word they are built on, and this physical picture is what the rest of the New Testament borrows when it speaks about suffering.