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I cannot give any reasons for the purpose of connections, as I myself have yet to study them. As for differential forms, I will give you a topological reason to study them. The topological motivation is de Rham cohomology. An inherently analytic object such as a differential $k$-form $\omega$, depending on whether or not it solves the equation $\omega = d\eta$, can help to determine the de Rham cohomology groups$$H^k_{dR}(M) := \frac {\ker(d:\Omega^k(M)\to\Omega^{k+1}(M))}{\operatorname{im} (d:\Omega^{k-1}(M)\to\Omega^k(M))}$$ which, as it turns out, are homotopy invariants (and by a theorem of de Rham himself, isomorphic to the singular cohomology with real coefficients $H^k(M;\mathbb R)$). Thus, by studying the solvability of certain differential equations involving forms, we can determine something about the topology of the underlying topological space. $$\\\\$$As for the Lie derivative, the real motivation is that we want to do differential calculus with vector/tensor fields. So, we take the flow of the vector field whose direction we want to differentiate in, pull back the vector field being differentiated, and take the derivative at $t=0$, to explain better, let $V$ and $W$ be vector fields on a smooth manifold $M$, with $\theta:M\times\mathbb R\supseteq\mathcal D\to M$ the flow of $V$. Because $\theta_t$ is a diffeomorphism from the set $\{p\in M| (t,p)\in \mathcal D\} := M_t$ to the set $M_{-t}$, with inverse $\theta_{-t}$, we can use it to pull back vector fields, as follows: define $(\theta_t^*W)_p$ by $$\theta{_{-t}*} W_{\theta_t(p)}\in T_pM.$$ We can then take the following limit$$\lim_{t\to 0}\frac{(\theta_t^*W)_p -W_p}t := \mathcal (L_VW)_p = \frac d{dt}\big |_{t= 0}(\theta_t^*W)_p$$ which we call the Lie derivative. What it allows us to do is to "transport" a vector field at one point to a vector at another, and find the rate of change of that transformation. Why do we do this? Because, it is a step along the way in our quest of generalizing analysis to an arbitrary smooth manifold. It allows us to take the derivative of a vector field "along" the flow of another. However, there is a problem. Lie derivatives do not allow us to take directional derivatives along curves, as it leads to problems. This is why we use the notion of a connection/covariant derivative (My source for this is the following Math.SE question: Link). As for your question on the tangent bundle, it allows us to study things globally, and we can use the machinery of connections, parallel transport, etc. Hope this helps. $$\\\\$$ My main source is John Lee's Introduction to Smooth Manifolds.