I feel this verse can be better understood if we look in the 2nd chapter of Acts. When the 120 people (Acts 1:15) in the upper room were filled with the Holy Spirit, they "began to speak in other toungues as the Spirit enabled them" (2:4). The crowd of "God-fearing Jews from every nation under Heaven" (v.5) that were gathered outside were amazed because they heard the 120 "declaring the wonders of God in [their] own toungues" (2:11). Not understanding how this could be possible and what it all meant, they thought the 120 were all drunk (2:13). Peter then went outside with the other disciples and gave what is arguably the best altar call since Jesus Christ Himself (2:14-41), leading "about three thousand" to Christ (2:41).
The gift of speaking in toungues - or praying in the Spirit, as some like to call it - is a sign to the unbeliever of the wonders and power of God. Many who do not believe in God through His Son, Jesus Christ, will feel that those speaking in tongues are "out of [their] minds" (1 Corinthians 14:23) and scoff like the God-fearing Jews did on the day of Pentecost (some believers still feel this way today, but that's a whole other subject). But if we read on in 1 Corinthians 14, we see that Paul tells us that when the speaking in toungues is taken to the next level - prophesying - the unbeliever "will be convinced by all that he is a sinner and will be judged by all, and the secrets of his heart will be laid bare. So he will fall down and worship God, exclaiming, "God is really among you!" (v.24b, 25). Therefore, the speaking of toungues is a way of getting an unbelievers attention and, when given to prophesying, turns the unbeliever's heart to God and can make him a believer in Jesus Christ.
I'm not sure if I have explained my understanding of this somewhat difficult concept very well. I have tried to put into words what understanding the Holy Spirit has given me. Since first reading your post, I prayed upon your question throughout the day yesterday and this morning. If I have confused you further, I do sincerely apologize.
One last statement I would like to make, and I feel Pastor Adrian Rogers of Bellevue Baptist Church and his Love Worth Finding ministry said this best, and I quote (paraphrase, rather), "Are the Gifts of the Holy Spirit operational in today's Church? I won't say 'Yes' and I won't say 'No.' To do so would be putting God in a box and I'm not about to do that, beloved." Only God can define Himself and He will give us each better understanding of Himself as He enables us through His Holy Spirit and as we meditate on the mystery of His Being.
Father, I pray now in the Mighty Name of Your Son, my Lord and Savior, Jesus Christ, that you will reveal more of Yourself in my life and the lives of others as we seek Your Face each day. Draw closer to us as we draw closer to You. Do not hide your face from us but look graciously toward us that we may know You more. Through Jesus I pray. Amen.