Loosely speaking, any sudden change in a signal is regarded as a transient, and transients in an input signal disturb the steady-state operation of a filter, resulting in a transient response at the filter output. This leads us to ask how do we define ``transient'' in a precise way? This turns out to be difficult in practice.
A mathematically convenient definition is as follows: A signal is said to contain a transient whenever its Fourier expansion  requires an infinite number of sinusoids. Conversely, any signal expressible as a finite number of sinusoids can be defined as a steady-state signal. Thus, waveform discontinuities are transients, as are discontinuities in the waveform slope, curvature, etc. Any fixed sum of sinusoids, on the other hand, is a steady-state signal.
In practical audio signal processing, defining transients is more difficult. In particular, since hearing is bandlimited, all audible signals are technically steady-state signals under the above definition. One way to pose the question is to ask which sounds should be ``stretched'' and which should be translated in time when a signal is ``slowed down''? In the case of speech, for example, short consonants would be considered transients, while vowels and sibilants such as ``ssss'' would be considered steady-state signals. Percussion hits are generally considered transients, as are the ``attacks'' of plucked and struck strings (such as piano). More generally, almost any ``attack'' is considered a transient, but a slow fade-in of a string section, e.g., might not be. In sum, musical discrimination between ``transient'' and ``steady state'' signals depends on our perception, and on our learned classifications of sounds. However, to first order, transient sounds can be defined practically as sudden ``wideband events'' in an otherwise steady-state signal. This is at least similar in spirit to the mathematical definition given above.
In summary, a filter transient response is caused by suddenly switching on a filter input signal, or otherwise disturbing a steady-state input signal away from its steady-state form. After the transient response has died out, we see the steady-state response, provided that the input signal itself is a steady-state signal (a fixed linear combination of sinusoids) and given that the filter is LTI.