What is yoniso manasikara and ayoniso manasikara? As suggested in this answer, it is a good idea to investigate this further.

What is wise reflection (yoniso manasikāra) and unwise reflection (ayoniso manasikāra)?

Yoniso manasikāra is also translated as wise attention, appropriate attention.

Why is yoniso manasikāra important? What is it used for? How does one practise it?
The way my teacher explained it to me, we can either be masters of our mind, masters or information (=wise attention), or our mind can be our master, information can be our master (=unwise attention).

In the second case, we let our minds push us around. We allow information to make us think about things whether it's good for us or not. Things that are negative, or things that are useless to think about, things that make us feel bad. Our mind tells us: "see this? Because this is so and so, you have lost, there's no hope", etc. We let our negative inertia maintain ourselves in maimed and injured condition. Or, "see this? you must want this." - as is the case of lust or envy. We eat all kinds of informational junkfood without being selective. Or we let our rational mind, our logical mind, take us down the path of conceptual proliferation and fruitless quasi-intellectual speculation. We get fooled by information into believing that we must come to certain conclusions, that we don't have a choice. 

Yoniso manasikāra is Emotional Intelligence. We decide what to think and how to think it - in order to feel stronger, healthier, and motivated. We actively decide what not to pay attention to, so negative mindstates cannot enter our minds. We don't let "Mara", devas, spirits, memes, advertisement, etc. and other informational viruses, to enter our informational space uncontrollably and do whatever they want to us. We are actively responsible for our mental hygiene.

We know the noble truths, so we know how attachment to expectation mismatch causes dukkha and makes us weak, and how being blameless and having no inner conflict makes us happy and strong, so we actively exploit this mechanism.

Moreover, we decide how to frame our thinking, what context and coordinates to choose. Because reality is an interpretation we make, choosing a mental framework is choosing a reality. We choose reality in which we are happy and can make others happy. We don't choose reality in which we are miserable and make others miserable.