Stationarity is a concept defined for stochastic processes. Since we look at the process as random function then we can extend the definition of stationarity to functions. I.e a stationary function is a realisation of stationary process. This extension is quite informal though.
Stationarity for a process means that the for a number of points $t_1,...,t_n$, $(X_{t_1},...,X_{t_n})\sim (X_{t_1+h},...,X_{t_n+h})$, where $\sim$ means equality in distribution. The question would be, what does this entail? For example take two points of the process which are close to each other $(X_t,X_{s})$. The relationship between these two points is expressed by their distribution function. Let us say the points are closely related, i.e. the correlation $corr(X_t,X_s)=0.9$. Then if we shift these points by any distance, due to stationarity their relationship remains the same, i.e. $corr(X_t,X_s)=corr(X_{t+h},X_{s+h})=0.9$. Informaly in this case we can say that $X_{s+h}$ would follow $X_{t+h}$ similar to the way $X_{s}$ follows $X_t$.
Now due the way stationarity is defined this holds for any number of points. So in some sense function should look similar at different locations. Note that this depends on the distributional properties of the process. White noise is a stationary process too and it does not look similar at different points in the usual common sense.
