You have to lift up your body mass under the presence of gravity, so you have to overcome the force of gravity.
While climbing up the stairs you have to put force on ground, more then your weight, which put same force but in opposite direction i.e. on you.
Suppose your mass is $m$, then while climbing up, if you resolve the forces then,
$$N-mg=ma\text{, } N=-F$$
$F$ is the force you apply on the stair and $N$ is the force exerted by the stair on you. Newtons' third law, or the magnitude of $N$ is equal to that of $F$
So in $y$-direction, if you want to move up then the $N$ should be more then your weight ($mg$) and according to this equation you will have some acceleration.
However when you are walking (moving along x-axis) $N$ goes equal to your weight ($mg$) and both balance each other along $y$-axis so you have to only care about exerting force to move along $x$-axis.
That's why walking up stairs harder than walking normally.