It means that two things are true.

First:

$$ P(X_1 < t) = P(X_2 < t) $$

for all real numbers $t$ (i.e., $X_1$ and $X_2$ have the same distribution, often the shorthand equidistributed is used to describe this condition).

Second:

$$ P(X_1 < t) = \frac{1}{\sigma \sqrt{2 \pi}}\int_{-\infty}^t e^{\frac{(t - \mu)^2}{2 \sigma^2}} $$

for some fixed numbers $\mu$ and $\sigma$ (i.e. the distribution of $X_1$ (*) is a normal distribution).

This doesn't imply that $(X_1, X_2)$ is joint normal without further assumptions. If that was intended, it's not what the author actually wrote.

(*) Given the first condition, this implies that the distribution of $X_2$ is also a normal distribution.

It means that two things are true.

First:

$$ P(X_1 < t) = P(X_2 < t) $$

for all real numbers $t$ (i.e., $X_1$ and $X_2$ have the same distribution, often the shorthand equidistributed is used to describe this condition).

Second:

$$ P(X_1 < t) = \frac{1}{\sigma \sqrt{2 \pi}}\int_{-\infty}^t e^{\frac{(x - \mu)^2}{2 \sigma^2}} \,\text{d}x$$

for some fixed numbers $\mu$ and $\sigma$ (i.e. the distribution of $X_1$ (*) is a normal distribution).

This doesn't imply that $(X_1, X_2)$ is joint normal without further assumptions. If that was intended, it's not what the author actually wrote.

(*) Given the first condition, this implies that the distribution of $X_2$ is also a normal distribution.