Generally speaking, "rounding" a number means mapping that number to its nearest neighbor in a smaller set of numbers. The goal is the minimize the error between the original number and the approximation obtained by rounding.
For example, if I want to round $\pi$ to the nearest integer, I might note that $\pi$ is between $3$ and $4$, and then further note that $\pi$ is closer to $3$ than to $4$: roughly speaking,
$$ |\pi - 3| \approx 0.14 \qquad\text{and}\qquad |\pi - 4| \approx 0.86, $$
where $|a-b|$ (the absolute value of the difference) gives the distance between two numbers. Since $0.14 < 0.86$, $\pi$ is rounded down to $3$.
Something similar occurs in the example given in the original question. The goal is to round a number to the nearest hundredth. We might immediately note that
$$1731.91 < 1731.9146 < 1731.92, $$
so it will get rounded to one of those two numbers. Then observe that
$$ |1731.9146 - 1731.91| = 0.0046 \qquad\text{and}\qquad |1731.9146 - 1731.92| = 0.0054. $$
Since $0.0046 < 0.0054$, the number is rounded to $1731.91$, as this minimizes the error between the original number and the approximation.
As to why it is sufficient to look only at the digit that is directly to the right of the desired level of precision, the following may be helpful:
To simplify the discussion, suppose that the goal is to round a number between $0$ and $1$ to either $0$ or $1$. The same idea can be extended to rounding numbers to the nearest integer, or to other decimal places. On the number line, these are the numbers
Any number $N < 1/2$ rounds down to $0$, and any number $N > 1/2$ rounds up to $1$ (if $N$ happens to be exactly $1/2$, we have to make a decision about how to round it—the usual convention is to always round $0.5$ up to $1$, but there are other conventions, e.g. banker's rounding).
Notice that if $N < 1/2$, then $N$ can be written in decimal form as
$$ 0.dxxx\dotso, $$
where $d$ (the digit in the tenths place) is less than $5$. The remaining digits farther to the right are irrelevant, and could be anything at all. Thus if the digit in the tenths place is less than $5$, round down.
On the other hand, if $N > 1/2$, then $N$ can be written in decimal form as
$$ 0.dxxx\dotso, $$
where $d$ is at least $5$ (greater than or equal to $5$). Thus if the digit in the tenths place is $5$ or greater, round up.
