In sequent calculus, Weakening (or Thinning) and Contraction are the so-called structural rules of inference.
Weakening rule introduces an extra formula $D$ in the antecedent: :
$$\frac{\Gamma\vdash \Delta}{D,\Gamma\vdash \Delta} \text {LW}$$
or in the succedent :
$$\frac{\Gamma\vdash \Delta}{\Gamma\vdash \Delta,D} \text {RW}$$
of the sequent $\Gamma \vdash \Delta$, where $\Gamma, \Delta$ are sets of formulae.
The "meaning" of the rule is the following :
if we have a derivation of the sequent $\Gamma \vdash \Delta$, we can "add" a formula $D$ to the set of assumptions .
Contraction rule is :
$$\frac{D,D,\Gamma\vdash \Delta}{D,\Gamma\vdash \Delta} \text {LC}$$
and :
$$\frac{\Gamma\vdash \Delta,D,D}{\Gamma\vdash \Delta,D} \text {RC}$$
The "meaning" of the rule is the following :
we can always "cancel" redundant occurrences of a formula in the antecedent or succedent of a sequent.
Added
The basic semantic definitions for sequents are :
a sequent $\Gamma \vdash \Delta$ is satisfied in an interpretation if either some formula in $\Gamma$ is not satisfied, or some formula in $\Delta$ is satisfied
and :
a sequent is valid if it is satisfied in every interpretation.
This means that we have to "read" a sequent as follows :
$$(\gamma_1 \land \ldots \gamma_n) \rightarrow (\delta_1 \lor \ldots \delta_m)$$
where $\Gamma = \{ \gamma_1, \ldots, \gamma_n \}$ and $\Delta = \{ \delta_1, \ldots, \delta_m \}$.
For simplicity, assume that $\Gamma = \{ \gamma_1, \gamma_2 \}$ and $\Delta = \{ \delta_1, \delta_2 \}$.
The semantic definition above says that the sequent is valid iff, for every interpretation, either some formula in the antecedent is false or some formula in the succedent is true, i.e. :
$$\vDash (\gamma_1 \land \gamma_2) \rightarrow (\delta_1 \lor \delta_2)$$
If it so, then we can add a formula $D$ whatever to the antecedent, and if the conjunction $\gamma_1 \land \gamma_2$ is false the new conjunction $(\gamma_1 \land \gamma_2) \land D$ will still be false or we can add a formula $D$ whatever to the succedent, and if the disjunction $\delta_1 \lor \delta_2$ is true the new disjunction $(\delta_1 \lor \delta_2) \lor D$ will still be true.
In both cases, if the upper sequent $\Gamma \vdash \Delta$ is valid, then also the lower ones : $D,\Gamma \vdash \Delta$ and $\Gamma \vdash \Delta,D$ are.