What are Lie groupoids intuitively?
Here is an expansion of my comment, by request. The 2-category of Lie groupoids $\mathrm{LieGpd}$ admits the category of manifolds $\mathrm{Mfld}$ as a full sub-2-category (i.e. $\mathrm{Mfld} \to \mathrm{LieGpd}$ induces an iso on hom-groupoids), but the category of Lie groups $\mathrm{LieGrp}$ only has a functor $\mathrm{LieGrp} \to \mathrm{LieGpd}$, not full on 2-arrows, sending a group to the coresponding one-object groupoid. I think this is one reason why Lie groupoids are not best seen as a generalisation of Lie groups.
Given a group $G$ acting on $M$, there is a groupoid $M//G$ with objects $M$, morphisms $M\times G$ with source and target given by projection and action, respectively. This includes the boring case where $M=*$, but the mix of potentially positive dimensional orbit space $M/G$, interesting orbits $mG$ nontrivial stabilisers $G_m$ is more typical of what is happening in the Lie groupoid case. Nice Lie groupoids, for instance proper Lie groupoids $X$, where $(s,t)\colon X_1\to X_0\times X_0$ is a proper map, locally look like group actions (times a codiscrete groupoid $U\times U \rightrightarrows U$), so this is sometimes a lot of what you need to consider anyway. Functors are just equivariant maps, where you can be equivariant with respect to a homomorphism $G\to H$.
Generalising from this to general Lie groupoids you need to forget the fact that the source map has identical fibres, and that there isn't some global group containing all the stabilisers.
In the same way as groupoids $(\mathrm{src},\mathrm{trg}):X_1\rightrightarrows X_0$ are a symultaneous generalization of group actions
$$(\mathrm{pr}_1,\mathrm{act}):G\times X\rightrightarrows X$$
$$(\mathrm{pr}_1|_R,\mathrm{pr}_2|_R):R\rightrightarrows X\;,\qquad R\subseteq X\times X$$
on sets $X$, Lie groupoids are a simultaneous generalization of (smooth) Lie group actions and "smooth" equivalence relations (where "smooth" means the maps induced on $R\subseteq M\times M$ by the projections to the first and second factor are submersions) on manifolds $M$.
Group actions and equivalence relations, hence also groupoids, should have quotients, but a quotient manifold does not always literally exist; a Morita morphism between two groupoids is intuitively a smooth morphism between the putative quotient manifolds. The geometric objects that replace the non existing quotient manifolds are called (differentiable) stacks.
Perhaps you should look through some of the papers on Ronnie Brown's website. In particular
http://www.groupoids.org.uk/pdffiles/bedlewopaper4bcclass.pdf
Lie groupoids came from Ehresmann's work on fibre bundles in diff. geom. and were taken forward by Pradines and his ideas on Monodromy and Holonomy groupoids (The exact references to those are in Ronnie Brown's article, and also in the reference below.)
Also of use would be Kirill Mackenzie's introduction
http://kchmackenzie.staff.shef.ac.uk/publications/front4web.pdf
and various parts of that book may help.
I hope this helps.