What is the importance of intention in Buddhism?

From the Nibbedhika Sutta:

"Intention, I tell you, is kamma. Intending, one does kamma by way of body, speech, & intellect.

Also from Cetana Sutta (SN 12.38) below, we see that intention (cetana) provides a landing point for the continuation of karmic consciousness.

[the Blessed One said,] "What one intends, what one arranges, and what one obsesses about: This is a support for the stationing of consciousness. There being a support, there is a landing [or: an establishing] of consciousness. When that consciousness lands and grows, there is the production of renewed becoming in the future. When there is the production of renewed becoming in the future, there is future birth, aging & death, sorrow, lamentation, pain, distress, & despair. Such is the origination of this entire mass of suffering & stress.

The story of blind elder monk Cakkhupala from Dhammapada 1 shows the importance of intention:

On one occasion, Thera Cakkhupala came to pay homage to the Buddha at the Jetavana monastery. One night, while pacing up and down in meditation, the thera accidentally stepped on some insects. In the morning, some bhikkhus visiting the thera found the dead insects. They thought ill of the thera and reported the matter to the Buddha. The Buddha asked them whether they had seen the thera killing the insects. When they answered in the negative, the Buddha said, "Just as you had not seen him killing, so also he had not seen those living insects. Besides, as the thera had already attained arahatship he could have no intention of killing and so was quite innocent."

Verse 1: All mental phenomena have mind as their forerunner; they have mind as their chief; they are mind-made. If one speaks or acts with an evil mind, 'dukkha' follows him just as the wheel follows the hoofprint of the ox that draws the cart.

Also the story of the hunter's wife from Dhammapada 124 shows the importance of intention:

The bhikkhus then asked the Buddha, "Venerable Sir, is the wife of the hunter who is a sotapanna, also not guilty of taking life, if she has been getting things like nets, bows and arrows for her husband when he goes out hunting?" To this question the Buddha answered, "Bhikkhus, the sotapannas do not kill, they do not wish others to get killed. The wife of the hunter was only obeying her husband in getting things for him. Just as the hand that has no wound is not affected by poison, so also, because she has no intention to do evil she is not doing any evil."

Verse 124: If there is no wound on the hand, one may handle poison; poison does not affect one who has no wound; there can be no evil for one who has no evil intention.

So, intention is kamma. You can see in this question that buying frozen meat from the supermarket does not incur the kamma of killing, because there was no intention or action to kill.

But does this mean that intention without acting upon the intention can result in kamma?

According to the Yodhajiva Sutta:

When a warrior strives & exerts himself in battle, his mind is already seized, debased, & misdirected by the thought: 'May these beings be struck down or slaughtered or annihilated or destroyed. May they not exist.' If others then strike him down & slay him while he is thus striving & exerting himself in battle, then with the breakup of the body, after death, he is reborn in the hell called the realm of those slain in battle.

Here, the warrior has the intention. But when he strives and exerts himself based on the intention, then the fruit of the kamma reaches its peak.

Despite this, thoughts are also kamma that can result in suffering. From Mahakammavibhanga Sutta:

After doing an intentional deed to be experienced as pleasant by way of body, speech, or mind, one feels pleasure. After doing an intentional deed to be experienced as painful by way of body, speech, or mind, one feels pain. After doing an intentional deed to be experienced as neutral by way of body, speech, or mind, one feels neutral.”

Also the Bhumija Sutta:

"When there is a body, pleasure & pain arise internally with bodily intention as the cause; or when there is speech, pleasure & pain arise internally with verbal intention as the cause; or when there is intellect, pleasure & pain arise internally with intellectual intention as the cause.

OP: so how do these teachings influence Buddhists?

You can see in this question that buying frozen meat from the supermarket does not incur the kamma of killing, because there was no intention or action to kill.

Another example is the case where if you give money to a beggar, he might use it to buy alcoholic beverages. Some people may have the idea that if the beggar uses the money to buy alcoholic drinks, then they would share in the kammic consequence. However, in this case, the kamma is based on the intention of the giver. If you gave the money with intention to help the beggar buy food, then the result will be good (assuming that the beggar hid his intentions from you).