For the purposes of intuition, let me write an "answer" which is probably close to the way Picard or Lefschetz would have thought about this. Suppose that you have a family of complex nonsingular plane cubics $X_t$ degenerating to a nodal cubic $X_0$. It is possible to understand the change in topology rather explicitly.

You can set up a basis of (real) curves $\alpha_t,\beta_t\in H_1(X_t,\mathbb{Z})$, so that $\beta_t\to \beta_0\in H_1(X_0)$ and $\alpha_t\to 0$ as $t\to 0$. So that $\alpha_t$ literally is a vanishing cycle. There is more. As you transport these cycles around a loop in the $t$-plane, you end up with a new basis $T(\alpha_t),T(\beta_t)$. This is related to the old basis by the Picard-Lefschetz formula $$T(\alpha_t)=\alpha_t$$ $$T(\beta_t) = \beta_t \pm (\alpha_t\cdot\beta_t)\alpha_t$$ You visualize this by cutting along $\alpha_t$ and giving a twist before regluing.

You can jazz this up in various ways of course, and this is the modern theory of vanishing cycles.

For the purposes of intuition, let me write an "answer" which is probably close to the way Picard or Lefschetz would have thought about this. Suppose that you have a family of complex nonsingular plane cubics $X_t$ degenerating to a nodal cubic $X_0$. It is possible to understand the change in topology rather explicitly.

You can set up a basis of (real) curves $\alpha_t,\beta_t\in H_1(X_t,\mathbb{Z})$, so that $\beta_t\to \beta_0\in H_1(X_0)$ and $\alpha_t\to 0$ as $t\to 0$. So that $\alpha_t$ literally is a vanishing cycle. There is more. As you transport these cycles around a loop in the $t$-plane, you end up with a new basis $T(\alpha_t),T(\beta_t)$. This is related to the old basis by the Picard-Lefschetz formula $$T(\alpha_t)=\alpha_t$$ $$T(\beta_t) = \beta_t \pm (\alpha_t\cdot\beta_t)\alpha_t$$ You visualize this by cutting along $\alpha_t$ and giving a twist before regluing.

You can jazz this up in various ways of course, and this is the modern theory of vanishing cycles. In modern notation you'll see a nearby cycle functor $R\Psi$, which corresponds roughly to $H^*(X_t) = H_*(X_t)^*$ and a vanishing cycle functor $R\Phi$ which measures the difference between $H^*(X_t)$ and $H^*(X_0)$.