Endogenous Growth in the case of externalities (Learning by doing) so for the Learning by doing model productivity is defined as $$A_t=BK_t$$.

Inserting this into the optimum conditions leads us to the following:$$r_t=\alpha BL^{1-\alpha}$$ and $$w_t=(1-\alpha)B^{1-\alpha} K_t/L$$.

As we can see, the interest rate here is constant and does not change with capital $K_t$. Intuitively, the reasoning is that more capital used in an individual firm reduces marginal product of capital. However, more capital of an individual firm level also leads to a higher aggregated capital stock (if all firms increase their stock) and hence also to higher productivity A, that can compensate for the decreasing marginal product of capital.

However, could someone explain to me why $w$ increases with $K_t$. What does this intuitively mean?
You don’t explain what $w$ even is but I will assume it’s wage. The wage increases with the capital accumulation for several reasons. First, as the capital becomes more abundant the labor inputs become relatively more scarce and hence more valuable. Thus, also the price of labor relatively to capital should increase through higher wages.

Second, in most commonly used production function capital needs to be used with labor for example with standard Cobb-Douglass production function $AK^{\alpha}L^{1-\alpha}$ if you only use capital and no labor output is 0 no matter how much capital you employ. Hence using these kind of production function imply that when the capital stock increases, demand for labor increases as well putting more pressure on wages.

Another way how to think about it more intuitively is to think about labor productivity. It’s well established that wages vary positively with labor productivity and working with more capital makes people more productive hence increasing their wages.