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Of course your "intuition" request can be only about Set-based category where limits and colimits are based on Set analogue.

ABout Colimit you can think as a "amalgamated" union like glueing for a descent data (see also Boubaky- Topology (I vol.)).

ABout limits, is different, limits belong to the prodoct (discrete limit), and for a "geometric" representation the dimention grow...

Anyway an elements of $Lim_{i\in I}X_i$ can be view as a coherent chain (of the some shape of $I$) of elements: just one $x_i$ for any space $X_i$, these are connected togheter by links i.e. diagram morphism maps, and the coherence means that these elements are mapped togheter by these maps.

Anyway an elements of $Lim_{i\in I}X_i$ can be view as a coherent chain of elements: just one $x_i$ for any space $X_i$, these are connected togheter by links i.e. diagram morphism maps, and the coherence means that these elements are mapped togheter by these maps.