As we begin to answer this question, it is good to know that the word "always" is a translation of three Greek words. A literal translation of those three words, en pantí kairoó, would be "in all times," and many Bible translations have chosen to use similar wording, such as "at all times." Other versions may use "all the time," while some use words like "constant" and "constantly." Weymouth's New Testament goes so far as to read, "every moment."
Christ is speaking, not just about prayer, but also about the frequency of our prayers. How often are we in contact with God throughout our day? Do we give Him some time in the morning or evening, but the rest of the day He is in none, or very few, of our thoughts? Doing so places us in very dangerous company (Psalm 10:4).
Laodiceans have lukewarm relationships with God, thus Christ has to say to them in Revelation 3:20: "Behold, I stand at the door and knock. If anyone hears My voice and opens the door, I will come in to him and dine with him, and he with Me." He calls for them to rekindle the relationship. Making the first move, He suggests what friends who have a close relationship do—they share a meal. What happens at a meal with friends? Conversation, which is what prayer is. Humans, whether with people or with God, build their relationships the same way: They talk to each other—a lot.
We can see why Christ tells those living at the end, when Laodiceanism reigns, that we have to overcome and pray always. Generally, the relationship between God and the Christian is weak and must be rebuilt, requiring considerable conversation, prayer, at all times of the day.
If we observed a marriage in which the husband and wife only mumbled to each other a little in the morning and/or a little at night, we would conclude that that relationship was in trouble. Our God who sees all knows the same thing when He experiences it.
How does a Christian "pray always"? In one of Herbert W. Armstrong's radio broadcasts on the book of Hebrews, he says, paraphrased, "You need to be in contact with God every hour!" I Thessalonians 5:17 instructs, "Pray without ceasing." Hebrews 13:15 urges us to offer prayer to God "continually." God's purpose for us requires a great deal of contact with Him.