The materialistic mindset invented language built around the key notion of objects. Objects are represented by nouns.
In contrast to that, the spiritual mindset or the mind-over-matter mindset came up with the concept of dharma as its primary building block. Dharmas are not nouns because they are not objects. Instead, dharmas are adjectives because they represent qualities.
In the mind-first view of existence, the primary building block is an experienced quality, a dharma.
Dharmas arise into existence, stay for some time, and then disappear. Most dharmas don't arise momentarily but gradually form from some combination of previous dharmas. Then they mutate or change, often weakening over time. Eventually they vane away or gradually decompose and become components for other dharmas.
All dharmas, practically speaking, with some rather theoretical exceptions, are transient. This is because, being nothing but qualities of combinations of other qualities they morph away out of existence as their underlying qualities morph away as their underlying qualities morph away and so on.
When Buddha describes meditation on arising and passing away of dharmas, he is referring to gradual formation, deformation, and eventual morphing out of existence of all qualities observable by the meditator.
This includes so-called external qualities, and so-called internal qualities. "External" qualities include shape, size, color, loudness, pitch, smell, temperature etc. - morphing in and out of existence either quickly or slowly. "Internal" qualities include discursive, emotional, somatic, conceptual, abstract, intuitive etc. characteristics, morphing in and out of existence. Qualities in both groups arise and vane due to forces out of our control (such as the time passing) as well as due to forces in our control (such as us moving our gaze or our attention around).
This is what's called "arising and passing away of dhammas".
Specifically in context of Satipatthana the external dhammas are analyzed by their sensory basis (visible, audible, tactile etc) and the internal dhammas are analyzed by their effect on one's practice. The most important dhammas to watch are the qualities of mind that help or harm meditation, such as sensuality, negativity, torpor, anxiety, uncertainty, as well as the positive qualities such as mindfulness, determination, enthusiasm, focus, calmness etc.
The most abstract method of contemplating the qualities is by observing the Noble Truths in action, happening in front of one's very eyes. "This is the quality of dukkha (i.e. ~painful aspect of some experience) arising, this is the underlying quality of craving, here the quality of craving has been abandoned, here the dukkha quality is no longer to be found."