If you flick a taught rope from both ends, you will see waves forming and moving.
A rising peak happens because the particles were flicked upwards by an upwards force by the person, and this upwards force propagates from particle to particle.
A lowering valley happens likewise due to a downwards initial force which propagates.
When waves meet, then both forces act on the same particles. If they are equal, then a particle feels an upwards and a downwards force that are equal at the same time. It thus doesn't move due to Newton's 1st law.
This same mechanism happens with sound waves where the moving media is air molecules.
In other types of waves, such as electromagnetic radiation travelling through space with no medium, we can think of the rising radiation wave as a positive value of the electric or magnetic field or the like. Two meeting waves add together, and if one is upwards and one is downwards, then we have a positive and a negative value added together, causing them to mathematically cancel out.