Suppose they could meet at some point $P(x_p,y_p)$, in other words there exists a point $P$ which belongs to both lines.
That is: the same point $P$ belongs to each of the two lines.
This means a pair of coordinates $(x_p, y_p)$ satisfies both equations: $$y_p = x_p \quad \text{and} \quad y_p = x_p+1.$$ But equality is transitive: if two values equal to a third one, then the two are equal to each other.
So we have: $$x_p = y_p = x_p+1,$$ which results in $$x_p = x_p+1.$$ This is not satisfied by any real value $x_p,$ because it would require $$0=1$$ which is false.

As a result, there is no such $x_p$ at which a point $P$ might exist belonging to both lines, hence lines do not have a common point—so they never meet.

Suppose they could meet at some point $P(x_p,y_p)$, in other words there exists a point $P$ which belongs to both lines.
That is: the same point $P$ belongs to each of the two lines.
This means a pair of its coordinates $(x_p, y_p)$ satisfies both equations: $$y_p = x_p \quad \text{and} \quad y_p = x_p+1.$$ But equality is transitive: if two values equal to a third one, then the two are equal to each other.
So we have: $$x_p = y_p = x_p+1,$$ which results in $$x_p = x_p+1.$$ This, however, is not satisfied by any real value $x_p,$ because it would require $$0=1$$ which is false.

As a result, there is no such $x_p$ at which a point $P$ might exist belonging to both lines, hence lines do not have a common point—so they never meet.