Because the rods are made of same material, material properties such as specific heat capacity (c), mass density (\rho) and coefficient of volume expansion (\gamma) are the same.
If Q is the heat entering a body of mass m, volume V, density \rho and specific heat capacity c, then the change in temperature of the body \Delta T is related to other quantities as -
Q = mc\Delta T = (\rho.V)c\Delta T; \qquad
Rearranging,
V\DeltaT = (\frac{Q}{\rho.c}) ...... (1)
Since \rho and c are material constants, the the product of volume and change in temperature (V\DeltaT) will be a constant, if Q is a constant.
For a temperature change of \Delta T the volume expansion is -
\Delta V = \gamma (V\DeltaT) ...... (2)
Let V_1 and V_2 be the volumes of the two rods and \Delta T_1 and \Delta T_2 are the change in their temperatures when they absorb the same heat Q. Then the change in their volumes are -
\DeltaV_1 = \gamma(V_1\DeltaT_1); \qquad \DeltaV_2 = \gamma (V_2\DeltaT_2)
But we know from (1): \quad V_1\DeltaT_1 = V_2\DeltaT_2
Therefore, \quad \DeltaV_1 = \DeltaV_2; \qquad \rightarrow \DeltaV_1 : \DeltaV_2 = 1:1.
