How to develop patience?

Since you are not a monk, I am giving you a solution from householders point of view.Commit to yourself that whatever you will do you will do it slowly. You will not rush to be successful. You will not rush to eat. You will not rush to prove your point.
That is one way.. Another way is to sit and meditate. Meditate as long as possible. Meditate on short breath.Meditate on long breath. Just sit and do nothing. Patience automatically develops when you realize there is nothing worth being impatient for... Buddha says this is suffering. Whatever you crave for is the origin of suffering. If you are serious follow the 8 fold path for the cessation of suffering. 8 fold path will automatically develop patience in you.

Finally , why does impatience arise ? The answer is simple. It is due to greed or aversion to something. Get rid of greed and aversion ... impatience will fade away.

What Buddha said on this?

From the Diamond Sutra:

I also remember Subhuti that during my five hundred previous lives I had used life after life to practice patience and to look upon my life humbly, as though I were a saint called upon to suffer humility.

Patience is a very important character trait that one need to develop if you wish to walk this path. To develop this quality of ‘Kanthi’(Patience), it needs to be developed together with the other qualities of adhittana (unwavering determination), ariya metta (loving kindness to all irrespective of all odds), nekkhamma (renunciation), and upekkha (equanimity).

Our inherent nature as humans is one of haste. Haste is a common characteristic inherent in most people when dealing with day to day affairs. People tend to accomplish most affairs with undue haste whether these particular affairs need or do not need urgent attention for completion. It is evident most of the time that people are not in a composed, relaxed mental frame to accomplish most of these tasks with patience. Even their day to day physical activities such as talking, eating, walking is also done with undue haste. This undue haste is an inborn characteristic of human beings.

A story from the Buddha’s time on haste would shed light on our present day busy, hectic lifestyles. According to this story, Bikkhuni Uttara, aged 120 years, was on her alms rounds one day when she noticed the Buddha, who was also on the alms rounds with the other monks, coming towards her direction. Bikkhuni Uttara, out of respect for the Buddha, stepped aside hurriedly to make way for the Buddha. While stepping aside hurriedly, she stepped on her robe by accident and fell on the ground. The Buddha who saw this incident came to her and treated her with the assistance of others and expounded a discourse, emphasising the importance of acting without undue haste in life. At the end of this discourse, Bhikkuni Uttara attained the fruit of stream entry (sotapanna).

Good householder, coming after the original questioner (hopefully not inpatient that leaded to abounding account),

Patient endurance:
the foremost austerity.
— Dhp 184

patient (kanti) is sayed to be the sibling of metta. So best treated like metta in action.

Ingratitude and wrong view are the main reasons of inpatient:

"And how is one made impure in three ways by mental action? There is the case where a certain person is covetous. He covets the belongings of others, thinking, 'O, that what belongs to others would be mine!' He bears ill will, corrupt in the resolves of his heart: 'May these beings be killed or cut apart or crushed or destroyed, or may they not exist at all!' He has wrong view, is warped in the way he sees things: 'There is nothing given, nothing offered, nothing sacrificed. There is no fruit or result of good or bad actions. There is no this world, no next world, no mother, no father, no spontaneously reborn beings; no brahmans or contemplatives who, faring rightly & practicing rightly, proclaim this world & the next after having directly known & realized it for themselves.' This is how one is made impure in three ways by mental action. Cunda Kammaraputta Sutta.

So straighten right view is most importand to overcome signs of aversion and ill-will, and it has to start by oneself, within: with ones reaction, feeling, bad states of mind.

Best way is always to ask oneself on a call by ones thought-committee: "Sure? Let's wait first and see if real and lasting."

An extended collection on training how to overcome ill-will can be found here: 2. Ill-will

And this are the things from outward which should be learned to endure:

"[4] And what are the fermentations to be abandoned by tolerating? There is the case where a monk, reflecting appropriately, endures. He tolerates cold, heat, hunger, & thirst; the touch of flies, mosquitoes, wind, sun, & reptiles; ill-spoken, unwelcome words & bodily feelings that, when they arise, are painful, racking, sharp, piercing, disagreeable, displeasing, & menacing to life. The fermentations, vexation, or fever that would arise if he were not to tolerate these things do not arise for him when he tolerates them. These are called the fermentations to be abandoned by tolerating. MN 2

Since patient is also a importand criteria for being prepared to receive teaching right, my person will end here for now.

A collection of Suttas on the parami (perfection) of Kanti here: Endurance, in the study guide on the Ten Perfections.

[Note that this isn't given for stacks, exchange, other world-binding trades but for escape from this round bound]

In my understanding, it comes back to a central Buddhist concept of "attachment".

We feel frustrated or angry, we get emotional about things, when we have a strong attachment or prejudice about how things are supposed to be.

These prejudices or preconceptions come from some strong experiences we have in childhood, or from some books we trust, people we highly respect etc. They also come from pain, when we did something and got in trouble. In most of these cases we have some basic formula in our head "this is bad" or "this is good" (why - because blah blah blah). And we are so convinced that this is correct, that we feel like we will never change our opinion in the entire life.

So when we see something or someone doing something that goes against this preconception or attachment, then we feel like they go against Truth (or the way we see it anyway). This makes us frustrated or angry or upset etc.

There are multiple ways to develop patience in Buddhism. On the basic level, we just try to calm down. We tell ourselves how bad it would be if we lose control and start yelling etc. But in my experience this does not work as soon as situation gets serious. We just forget everything and jump right back into emotion.

The next method is to be mindful of one's emotions on the body level. This way we can notice ourselves getting nervous far before it gets out of control. So if we develop feeling of presence of body, and always feel all tensions etc., we get this convenient weather indicator that can show a storm coming 5 days in advance (really maybe 5 minutes, but still good). It is much easier to fix e.g. a conversation that goes towards a scandal 5 minutes before it gets out of control, when things are still calm, but you feel it coming.

And then the most powerful method is to see the attachments or preconceptions that give rise to emotion - and learn to let them go. This is the most difficult, because it means you have to learn to see situation from another perspective, and to not assume that you are 100% right. This can be very difficult in the beginning. But at the end this ability makes you practically immune to all disturbances, 100% patient at all times. It is a very helpful feature to have in life.

My teacher said, enlightened person is like a lawyer. He or she sees every situation from all sides at the same time. This helps him to never lose patience and to make better decisions than anyone. The point is, Buddhist patience is not self-restraint and self-torture, it is freedom from single point of view, freedom from attachments.

P.S.

Same exact logic applies to impatience towards desirable objects. Attachment, preconception etc. - except in this case it's a preconception that the object is "good", usually based on some overgeneralization or idealisation. So the method Buddha prescribed for that is to analyze the object and see that it has negative sides and not only positive. This helps to let go of attachment and come back to patience and balance. All the same advice about being mindful of your body to sense the attachment getting out of control, applies here as well. In Buddhism desire and aversion are considered things of the same nature, one is positive and the other is negative but the principle is the same.