Consider the SVD of the $N\times N$ matrix A to be,
$$A = U\Sigma V^T$$
where U and V are orthogonal matrices, and $\Sigma$ is a diagonal matrix with entries $$\sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_N \geq 0$$ For a ill-conditioned matrix $A$, all the singular values decay gradually to zero and the condition number, i.e.
$cond(A)= \frac{\sigma_1}{\sigma_N}$ is very large.
The SVD approach (a.k.a spectral filtering) is to damp the effects caused by division by the small singular values.
$$x_{naive} = A^{-1}b= V\Sigma^{-1}U^Tb = \sum_{i=1}^{N}\frac{u_i^Tb}{\sigma_i}v_i $$
The TSVD method is an example of the general class of methods that are called spectral filtering methods, which have form,
$$x_{filt} = \sum_{i=1}^{N}\phi_i\frac{u_i^Tb}{\sigma_i}v_i $$
where the filter factors $\phi_i$ are chosen such that $\phi_i \approx 1$ for large singular values, and $\phi_i \approx 0$ for small singular values.
Now look at the filter equations,
$\bf{The~TSVD~Method}$
$$\phi_i = 1, i = 1,\cdots, k $$
$$= 0, otherwise$$
The parameter $k < N$ is called the truncation parameter.
$\bf{The~Tikhonov~Method}$
$$\phi_i = \frac{\sigma_i^2}{\sigma_i^2 + \alpha^2}, i = 1,\cdots, N $$
The parameter $\alpha > 0$ is called the regularization parameter. This choice of filter factors yields the solution vector $x_{\alpha}$ for the minimization problem,
$min_x { ||b-Ax||^2_2 + \alpha^2 ||x||_2^2}$.
To answer your question, "when Tikhonov regularization becomes similar(or equal) to TSVD", we can see that as $\alpha \rightarrow 0$, $\phi_i \rightarrow 1$ which are the filter coefficients, and the Tikhonov method becomes similar to TSVD. You can think of this filtering as, TSVD uses a filter with a sharp jump from 0 to 1 and Tikhonov using a smoother approach (it does prevent oscillations to the solution.) For more details see Spectra and Filtering.
