Here is an example of a very hand-on (and standard) construction of the path that you're looking for. The goal will be take a sequence of diffeomorphisms in the path that are located sufficiently close to each other in the $C^\infty$ topology, and then to join these diffeomorphisms by smooth paths.

Choose an auxiliary Riemannian metric on $M$, and consider the "exponential map" $E \colon TM \to M \times M$ given by $v \mapsto (p(v),\exp_{p(v)}(v))$ where $p \colon TM \to M$ is the tangent bundle.

For a diffeomorphism $\varphi$ of $M$ we write $\Gamma_\varphi:=\{(x,\varphi(x))\} \subset M \times M$ for its graph (a smooth submanifold). Given that a diffeomorphism $\varphi$ is sufficiently $C^\infty$-close to the diagonal $\Gamma_{\mathrm{id}_M} :=\{(x,x); x \in M \} \subset M \times M$, it follows that $E^{-1}(\Gamma_\varphi) \subset TM$ consists of a smooth section $\zeta$ of $TM \to M$ together with possibly additional loci contained outside of some a priori fixed tubular neighbourhood of the zero-section. (Here I am admitting some applications of standard transversality results and properties of the exponential map.)

In other words $x \mapsto \exp_{x}(t\zeta)$ is the sought smooth path parametrised by $t$ which connects $\mathrm{id}_M$ (at $t=0$) and $\varphi$ (at $t=1$).