the branching-process heuristic
The branching-process heuristic is the single most important intuition in random graph theory: the local neighbourhood of a vertex in a sparse random graph looks like a Galton-Watson tree. It explains, in one stroke, why the giant component appears exactly at average degree 1, what the survival fraction is, and how component sizes are distributed. The question it answers is: why should a graph have a phase transition governed by the simple rule 'mean offspring greater than 1'?
Make it precise by exploring a component vertex by vertex. Start at a vertex v and reveal its neighbours; these are its children. Then reveal the new neighbours of each child not already seen; these are grandchildren; and so on (breadth-first or depth-first). In G(n, c/n), when we have explored only o(n) vertices, each unexplored vertex is still a neighbour of the current vertex independently with probability about c/n, so the number of new neighbours is approximately Binomial(n - (explored), c/n), which converges to a Poisson(c) law. Thus the exploration is approximately a Galton-Watson branching process with Poisson(c) offspring, as long as the explored set is small compared to n. A branching process with mean offspring c goes extinct with probability 1 if c <= 1 and survives with positive probability rho(c) > 0 if c > 1, where rho solves rho = 1 - e^(-c rho). Extinction means the component is finite (small); survival corresponds to the exploration running on for a linear time, i.e. hitting the giant. The approximation breaks once a constant fraction of vertices is explored — the 'depletion of points' — which is exactly what stops the giant from swallowing everyone and pins its size at rho(c) n.
This heuristic is the engine behind essentially every sparse-random-graph result: the giant's size, the small-component size distribution (which is the total-progeny law of the branching process), the number of components, the local structure (which converges to a Galton-Watson tree in the Benjamini-Schramm sense), and the analysis of epidemics and bootstrap percolation. It matters because it reduces a complicated dependent structure to an exactly solvable independent one. The honesty to keep: it is only a local approximation, valid while the explored set is o(n); the correction from depletion is essential and is what makes the giant finite, and at criticality (c = 1) the naive branching picture must be refined to the n^(2/3) and Brownian-excursion scaling because the critical Galton-Watson tree is barely-subcritical-or-critical and its total progeny has heavy (power-law, exponent 3/2) tails.
To find the giant fraction at c = 2: the cluster of v survives (is giant) with probability rho solving rho = 1 - e^(-2 rho), about 0.797. Equivalently, the extinction probability q = 1 - rho is the smallest root of q = e^(-2(1-q)) = G(q) where G(s) = e^(2(s-1)) is the Poisson(2) offspring generating function — the standard branching-process extinction equation.
Cluster exploration = a Poisson(c) Galton-Watson process; the giant exists iff that process can survive, i.e. iff c > 1.
The branching picture is valid only locally (while o(n) vertices are explored); ignoring depletion would wrongly let the giant cover everyone. At criticality the naive heuristic fails and must be replaced by the n^(2/3) / Brownian-excursion scaling.