a simple predictable integrand
The Ito integral is built in two stages, and a simple predictable integrand is the explicit, hands-on starting class on which the integral is defined directly, before any limiting argument. The idea is the same as in Lebesgue integration, where you first integrate simple (step) functions where the answer is obvious, then extend to general functions by approximation. Here the 'step functions in time, with random heights' are the simple predictable processes, and their integral against Brownian motion is just a finite sum of increments — no analysis required.
A process H is a simple predictable integrand if there is a finite partition 0 = t_0 < t_1 < ... < t_n = T and random variables xi_0, xi_1, ..., xi_{n-1} such that H_s = xi_k for s in the interval (t_k, t_{k+1}], where crucially each xi_k is F_{t_k}-measurable — that is, each height is known at the LEFT endpoint of its interval, using only information available strictly before the increment it will multiply. (More generally one writes H = xi_0 * 1_{{0}} + sum_k xi_k * 1_{(t_k, t_{k+1}]}, and predictability is exactly the requirement that the value be left-continuous / known in advance.) For such H the stochastic integral is defined by the unambiguous formula integral_0^T H dB = sum_k xi_k (B_{t_{k+1}} - B_{t_k}): each random height multiplies the Brownian increment over its own interval, and you add them up. This is a genuine Riemann-type sum, but with the evaluation point pinned at the left end, which is what makes the construction work.
Two facts make this class the right launching pad. First, on simple integrands the integral is manifestly a martingale (each increment is a centered increment of B, independent of the F_{t_k}-measurable height) and the Ito isometry is a one-line computation. Second, the simple predictable processes are DENSE in the space L^2 of all predictable square-integrable integrands; combined with the isometry, density lets the integral extend uniquely to the full space. The honest subtlety: predictability (left endpoint, height known in advance) is not optional bookkeeping — if you allowed the height xi_k to peek at the increment (say, evaluate at the right endpoint or midpoint), the integral would no longer be a martingale and the clean isometry would fail. That single modelling choice is the dividing line between the Ito and Stratonovich theories.
A gambler holds xi_k = sign(B_{t_k}) shares over (t_k, t_{k+1}] — betting on the direction Brownian motion currently points, a decision made at the left endpoint and hence F_{t_k}-measurable. Her gain is integral H dB = sum sign(B_{t_k}) (B_{t_{k+1}} - B_{t_k}), a legitimate simple-integrand Ito integral. If she could instead choose xi_k = sign(B_{t_{k+1}}) — betting on where B will be — she would be peeking into the future, the predictability would fail, and the 'integral' would no longer be a fair game.
Heights known at the left endpoint give a martingale (fair game); peeking at the right endpoint breaks predictability and the whole theory.
Predictability is stronger than adaptedness for a general process, but for simple integrands it just means each height is measurable with respect to the left endpoint of its time interval; this 'no peeking' rule is the entire reason the left-endpoint Ito convention is a martingale.