Why does $(-1) \times (-1)$ give +1? Why is $(-1) \times  (-1)=+1$ ?  What is the intuitive concept ?
 My second question :

How can I show that no triangular number can be of the form $3n-1$ ?
First, let's establish a pattern that can be used to intuitively understand why $1 \times (-1) = (-1)$

$1 \times 3 = 3$

$1 \times 2 = 2$

$1 \times 1 = 1$

$1 \times 0 = 0$

Every time the factor goes down by one, the product goes down by 1 (which is the first factor here).

Extending the pattern:

$1 \times (-1) = -1$

$1 \times (-2) = -2$

and so on.  So $a \times (-b) = - (a \times b)$

Well, we can apply the pattern to the other factor as well:

$(-1) \times 3 = -3$

$(-1) \times 2 = -2$

$(-1) \times 1 = -1$

$(-1) \times 0 = 0$

Every time the factor goes down by one, the product goes down by -1 (which is the first factor here), or in other words, goes up by 1.

Extending the pattern:

$(-1) \times (-1) = 1$

$(-1) \times (-2) = 2$

and so on.  So $(-a) \times (-b) = a \times b$

Now for the second question:

A triangular number is of the form $n(n + 1)/2$.  And of the form $3n - 1$ really means $-1 \pmod 3 \equiv 2 \pmod 3$.

If $n \equiv 0 \pmod 3$, $n(n + 1)/2 \equiv 0 \pmod 3$.

If $n \equiv 1 \pmod 3$, $n(n + 1)/2 \equiv 2/2 \equiv 1 \pmod 3$.

If $n \equiv 2 \pmod 3$, $n(n + 1)/2 \equiv 0 \pmod 3$.

None are equivalent to $2 \pmod 3$, so it can't be expressed as $3n - 1$.
$$(-1) \times (2) = -2$$
$$(-1) \times (1) = -1$$
$$(-1) \times (0) = 0$$
$$(-1) \times (-1) = \cdots$$

Of course, maybe some of these aren't intuitive either!