How do we understand these two Ahadeeth?
Dear noble brothers, the issue of infection was made clear by the Prophet PBUH more than 14 centuries ago. The following is a Hadith reported by al-Bukhari: 

(There is no such thing as infection…)

[Al-Bukhari]

In the above Hadith, Allah's Messenger PBUH who never  speaks out of his own desire, as all that he says is an inspiration inspired to him confirmed: 

(There is no such thing as infection…)

[Al-Bukhari]

However, infection is something that he himself confirmed in the following Hadith: 

(Usamah bin Zaid, may Allah be pleased with him, said: The Prophet  PBUH said, "If you get wind of the outbreak of plague in a land, do not enter it; and if it breaks out in a land in which you are, do not leave it.")

[Al-Bukhari and Muslim]

Why did Allah's Messenger PBUH command us not to enter, or get out of, a place afflicted with a plague? The answer is: for fear of getting infected. Well, most recent bacteriological studies confirm that there is no such thing as infection. This is because germs or viruses may be transferred from an ill person to a healthy one, yet the healthy person does not get infected. In other words, a healthy person may transfer a certain disease but does not get infected with it. Instead, he/she acquires more immunity against it, i.e. becomes more immune from such a disease. Furthermore, a certain disease may sometimes be transferred by a number of healthy people who bear it but do not get infected with it, and then, another healthy person, suddenly and unexpectedly, gets infected with it.

Hence, when the Prophet PBUH said that "There is no such thing as infection",  he meant that infection does not cause illnesses, but it provides a suitable environment for getting infected with such illnesses. This is most accurate!  

I would like to repeat the Hadith as a whole: 

(Narrated Abu Hurairah: Allah's Messenger PBUH said, "(There is) no 'Adwa (no contagious disease is conveyed without Allah's permission). nor is there any bad omen (from birds), nor is there any Hamah, nor is there any bad omen in the month of Safar, and one should run away from the leper as one runs away from a lion.")

[Al-Bukhari]

The Messenger's command in this Hadith to fly away from a leper does not at all contradict his reiteration that there is no such thing as infection. This is because infection does not cause an illness, but rather it provides a suitable environment for getting infected with it. 

A question and an answer:

A bacteriologist was once asked, "which parts of the body allow infection most?" "The four limbs and the face." He replied. Here we have to remember the  Wudu (Islamic ritual ablution) in which Allah's Messenger PBUH taught us not to wash our hands in the vessel but rather to pour water from the vessel to wash our hands outside it. This great Prophetic guidance prevents bacteria from getting into our bodies through our noses, mouths, ears, and hands. Allah's Messenger PBUH, commanded us to wash these parts at least five times a day. This shows clearly that Wudu provides utmost protection from getting infected with diseases, as it includes such hygienic procedures as rinsing the mouth and nose and washing the face, the hands and feet with pure clean water.

Prophetic Ahadeeth about infection:

Dear believing brothers, infection can also take place by means of the respiratory system, and this explains why the Prophet PBUH bade one of his noble Companions while he was drinking water:

(...'Remove the cup from your mouth and then breathe.' He said, 'Sometimes I see something floating in it?' He said, 'Then pour it out.'")

[Ahmad, on the authority of Abi Sa'id al-Khidri] 

In another Hadith, the Prophet PBUH said:

("Do not put a patient with a healthy person ). " (as a precaution).)

[Al-Bukhair, on the authority of Abi Hurairah]

The Hadith means that an ill person should not get in touch with a healthy one. Also, the healthy person should eat food offered to him in the patient's house as it might be infected.  

Pay attention to the following Hadith, please: 

(The Messenger of Allah (PBUH) as saying : There is no infection, no evil, omen or serpent, in a hungry belly and no hamah. A nomadic Arab asked: How is it that when camels are in the sand as if they were gazelles and a mangy camel comes among them and it gives them mange ? He replied: Who infected the first one? Ma'mar, quoting al-Zuhri said: A man told me that Abu Hurairah narrated to him saying that he heard the Prophet (PBUH) say: A diseased camel should not be brought with a healthy camel to drink water. He said: The man then consulted him and said: Did you not tell us that Prophet (PBUH) had said: There is no infection, no serpent in a hungry belly and no hamah? He replied: I did not transmit it to you. Al-Zuhri said: Abu Salamah said: He had narrated it and I did not hear that Abu Hurairah had ever forgotten any tradition except this one.)

[Sunan Abi Dawud, on the authority of Abi Hurairah]

The Prophet PBUH bade us to take precautions against diseases, but infection does not, alone, cause illnesses, as there are lots of other elements and causes that only the Almighty knows. Prophetic guidance is most accurate with this respect, as Allah's Messenger does not speak out of his own desire; and what he says is definitely the Maker's Divine Revelation and Commands.