I believe that an understanding of the taxation system can help to understand our place in it, and how that impacts each of our personal finances. I will try to remain unbiased here but this is a somewhat subjective question, so please bear with me if you disagree on any point.
Tax savings can occur for many reasons.
Some of these tax savings are well-advertised, and can be used by many people, such as tax credits for mass-transit passes which exists in some countries. But some of these tax savings are things you never heard of before, until it winds up on the news.
Why do some people seem to get tax savings that you and I cannot get, and why do those people always seem to have so much more money than us? A simplistic answer can show this in three parts: (1) The source of one's income; (2) Transaction costs; and (3) "tax loopholes".
1. Source of one's income
Tax savings occur proportionately to one's income, and if the savings apply to investment income, they occur proportionately to one's wealth. If someone living paycheck to paycheck with a minimal amount in a bank account "saves tax on investment income", they might reduce their taxable interest from $50 to $0. That's because they simply don't have any other investment income to reduce. All of their income comes in the form of employment, which is typically very hard to save taxes on. Most governments have a very firm grasp on the taxation of employment income, because it is a huge proportion of income in the country (and therefore has the largest amount of tax associated), and because it is very straightforward (work for someone = employment income). A more cynical person than I might point out that investment income is earned by the very wealthy, who can afford to lobby for politicians to pass favourable investment income laws.
2. Transaction costs
Even very straightforward tax saving opportunities may cost money to enable. The simplest example would be: if a tax saving opportunity is so complicated that an average person can't understand it themselves, then an accountant, lawyer, or banker will need to be the one to explain it. And that can cost you money. If your tax isn't so much to begin with, then the transaction costs to achieve the tax savings could be higher than the tax savings themselves. For example, most countries have tax savings / deferrals if you start a corporation. These rules typically exist to promote investment in the local economy. But someone who earns $10k in a side-business might not be able to afford the $3k in incorporation costs just to save $2k in taxes. The more income and wealth you have, the more these transaction costs become worthwhile.
3. "Tax Loopholes"
I'm going to generally define "tax loopholes" for the purposes of this answer as something where a somewhat arbitrary situation allows for taxes that a layman would consider unfair or unexpected. This often occurs with good intentions but poor legislation - the government tries to provide a benefit to a deserving group or to promote an activity, but ends up allowing another group to take advantage.
For example in Canada, there existed until a few years ago tax saving rules about passing on wealth to children at lower tax rates, only when a close family member is near-death [setting up a 'testamentary trust' between a grandparent and a grandchild could in some circumstances allow that trust to be created with additional 'tax brackets', meaning more income would be taxed at a less-than top tax rate before being distributed to the grandchildren].
The rules were put in place with the idea that "oh gee, a family member has died, and the dang ol' family is grieving so hard they can't distribute the wealth to the next generation for a few months on account of all the crying. We should make it so that the estate is taxed like a person, and if they earn only a little income, they have a low tax rate, and they only get taxed at the full rate if they have a lot of income". Seems reasonable enough, but if a family is ready to pass on wealth at the same time as someone is nudging the bucket with their foot, a morbid discussion with your lawyer and accountant could set your children up for life with forever reduced taxes on massive inheritances.
Panama / Paradise leaks
In the case of the Panama / Paradise leaks, tax savings are due to all 3 of the above: Those who have massive wealth (and therefore earn the majority of their income from investments instead of employment) can afford the transaction costs associated with taking advantage of specific "tax loopholes". The simplest example of which is just that income earned in a foreign country might have a lower tax rate than income earned domestically. This is often a result of "cracks" in the foreign tax treaties between countries, which exist generally to promote business between countries and prevent double-taxing individuals who need activity in both countries for whatever reason.
Take for example the "Apple loophole". Apple has operations around the world. Some activity occurs in low-tax jurisdictions. Apple reports a high percentage of the value of R&D as being associated with those jurisdictions. Those branches in low-tax jurisdictions charge the high-tax branches (such as the US) with fees for use of their valuable research. So much of Apple's income is reported in those foreign jurisdictions. It won't be taxed in the US until Apple "repatriates" the cash back to the US. Until then, the cash sits in the foreign jurisdiction, accruing less tax. This and similar rules can be used by individuals wealthy enough to hold corporations in foreign jurisdictions with low tax rates. How each particular rule / "loophole" works will depend on the nature of a specific case - tax law is complex, and the rules between countries are even more so.
These foreign tax loopholes are closing every year. It is getting harder and harder to hide money offshore, and it is getting less and less likely that you will be able to find a country with juuuust the right loopholes for your own offshore wealth. These types of news leaks will only help to expedite those changes.