For the martingale part, this follows by the Ito construction increments $B_{t_{i}}-B_{t_{i-1}}$. That implies that the sum of them is also a martingale $I(f^{n}) = \int_{0}^{t} f^{n}_u dB_u$ where $f^{n}$ are simple processes.
Then finally we use that the $L_1$-limit of martingales is still martingale eg. see here When is the limit of Martingales a Martingale?
To get the formulas for the moments we use the Gaussianity. Since $f$ is deterministic, we get that $X_{t}$ is Gaussian with variance $\mathbb{E}\int_0^t f(s)^2 ds$. see here
The integral of a progressively measurable process $f$ is a limit of the integrals $I(f^{n}) = \int_{0}^{t} f^{n}_u dB_u$ where $f^{n}$ are simple processes and these integrals are Gaussian by definition.
Then as mentioned there we just use
Let $X_n$ be a sequence of normally distributed random variables with mean zero and variances $\sigma_n^2 \in [0,\infty)$. Suppose $X_n \to X$ in distribution. Then $\sigma_n^2$ converges to some $\sigma^2 \in [0,\infty)$, and $X$ is normally distributed with mean zero and variance $\sigma^2$.
In fact, this process is just a time changed Brownian motion see here